Semimartingales and stochastic integration
Spring 2011
Sergio Pulido∗
Chris Almost†
Contents
Contents
1
0 Motivation
3
1 Preliminaries
1.1 Review of stochastic processes . . . . . . . . . . . . . . . .
1.2 Review of martingales . . . . . . . . . . . . . . . . . . . . .
1.3 Poisson process and Brownian motion . . . . . . . . . . .
1.4 Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Lévy measures . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Integration with respect to processes of finite variation .
1.8 Naïve stochastic integration is impossible . . . . . . . . .
∗
†
[email protected]
[email protected]
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Contents
2 Semimartingales and stochastic integration
2.1 Introduction . . . . . . . . . . . . . . . . . . . .
2.2 Stability properties of semimartingales . . . .
2.3 Elementary examples of semimartingales . .
2.4 The stochastic integral as a process . . . . . .
2.5 Properties of the stochastic integral . . . . . .
2.6 The quadratic variation of a semimartingale
2.7 Itô’s formula . . . . . . . . . . . . . . . . . . . .
2.8 Applications of Itô’s formula . . . . . . . . . .
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24
24
25
26
27
29
31
35
40
3 The Bichteler-Dellacherie Theorem and its connexions to arbitrage
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Proofs of Theorems 3.1.7 and 3.1.8 . . . . . . . . . . . . . . . . . . . .
3.3 A short proof of the Doob-Meyer theorem . . . . . . . . . . . . . . . .
3.4 Fundamental theorem of local martingales . . . . . . . . . . . . . . .
3.5 Quasimartingales, compensators, and the fundamental theorem of
local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Special semimartingales and another decomposition theorem for
local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
43
45
52
54
4 General stochastic integration
4.1 Stochastic integrals with respect to predictable processes . . . . . .
65
65
Index
72
56
58
61
Chapter 0
Motivation
Why stochastic integration with respect to semimartingales with jumps?
◦ To model “unpredictable” events (e.g. default times in credit risk theory)
one needs to consider models with jumps.
◦ A lot of interesting stochastic processes jump, e.g. Poisson process, Lévy
processes.
This course will closely follow the textbook, Stochastic integration and differential
equations by Philip E. Protter, second edition. We will not cover every chapter, and
some proofs given in the course will differ from those in the text. The following
numbers correspond to sections in the textbook.
I Preliminaries.
1. Filtrations, stochastic processes, stopping times, path regularity, “functional” monotone class theorem, optional σ-algebra.
2. Martingales.
3. Poisson processes, Brownian motion.
4. Lévy processes.
6. Localization procedure for stochastic processes.
7. Stieltjes integration
8. Impossibility of naïve stochastic integration (via the Banach-Steinhaus
theorem).
II Semimartingales and stochastic integrals.
1–3. Definition of the stochastic integral with respect to processes in L.
5. Properties of the stochastic integral.
6. Quadratic variation.
7. Itô’s formula.
8. Stochastic exponential and Lévy’s characterization theorem.
III Bichteler-Dellacherie theorem.
(NFLVR) implies S is a semimartingale
(NFLVR) and little investment if and only if S is a semimartingale
IV Stochastic integration with respect to predictable processes and martingale
representation theorems (i.e. market completeness).
For more information on the history of the development of stochastic integration, see the paper by Protter and Jarrow on that topic.
3
Chapter 1
Preliminaries
1.1
Review of stochastic processes
The standard setup we will use is that of a complete probability space (Ω, F , P)
and a filtration F = (F t )0≤t≤∞ of sub-σ-algebras of F . The filtration can be
thought of as the flow of information. Expectation E will always be with respect
to P unless stated otherwise.
Notation. We will use the convection that t, s, and u will always be real variables,
not including ∞ unless it is explicitly mentioned, e.g. {t|t ≥ 0} = [0, ∞). On the
other hand, n and k will always be integers, e.g. {n : n ≥ 0} = {0, 1, 2, 3, . . . } =: N.
1.1.1 Definition. (Ω, F , F, P) satisfies the usual conditions if
(i) F0 contains all the P-null sets. T
(ii) F is right continuous, i.e. F t = s<t Fs .
1.1.2 Definition. A stopping time is a random time, i.e. a measurable function
T : Ω → [0, ∞], such that {T ≤ t} ∈ F t for all t.
1.1.3 Theorem. The following are equivalent.
(i) T is a stopping time.
(ii) {T < t} ∈ F t for all t > 0.
PROOF: Assume T is a stopping time. Then for any t > 0,
[
_
{T < t} =
{T ≤ t − 1n } ∈
Fs ⊆ F t .
s<t
n≥1
Conversely, since the filtration is assumed to be right continuous,
\
\
{T ≤ t} =
{T < t + 1n } ∈
Fs = F t+ = F t ,
s>t
n≥1
so T is a stopping time.
ƒ
4
1.1. Review of stochastic processes
5
1.1.4 Theorem.
W
V
(i) If (Tn )n≥1 is a sequence of stopping times then n Tn and n Tn are stopping
times.
(ii) If T and S are stopping times then T + S is a stopping time.
1.1.5 Exercises.
(i) If T ≥ S then is T − S a stopping time?
(ii) For which constants α is αT a stopping time?
SOLUTION: Clearly αT need not be a stopping time if α < 1. If α ≥ 1 then, for any
t ≥ 0, t/α ≤ t so {αT ≤ t} = {T ≤ t/α} ∈ F t/α ⊆ F t and αT is a stopping time.
Let H be a stopping time for which H/2 is not a stopping time (e.g. the first
hitting time of Brownian motion at the level 1). Take T = 3H/2 and S = H, both
stopping times, and note that T − S = H/2 is not a stopping time.
ú
1.1.6 Definition. Let T be a stopping time. The σ-algebras of events before time
T and events strictly before time T are
(i) F T = {A ∈ F : A ∩ {T ≤ t} ∈ F t for all t}.
(ii) F T − = F0 ∨ σ{A ∩ {t < T } : t ∈ [0, ∞), A ∈ F t }.
1.1.7 Definition.
(i) A stochastic process X is a collection of Rd -valued r.v.’s, (X t )0≤t<∞ . A stochastic process may also be thought of as a function X : Ω × [0, ∞) → Rd or as a
random element of a space of paths.
(ii) X is adapted if X t ∈ F t for all t.
(iii) X is càdlàg (X t (ω)) t≥0 has left limits and is right continuous for almost all
ω. X is càglàd if instead the paths have right limits and are left continuous.
(iv) X is a modification of Y if P[X t 6= Yt ] = 0 for all t. X is indistinguishable
from Y if P[X t 6= Yt for some t] = 0.
(v) X − := (X t− ) t≥0 where X 0− := 0 and X t− := lims↑t X s .
(vi) For a càdlàg process X , ∆X := (∆X t ) t≥0 is the process of jumps of X , where
∆X t := X t − X t− .
1.1.8 Theorem.
(i) If X is a modification of Y and X and Y are left- (right-) continuous then
they are indistinguishable.
(ii) If Λ ⊆ Rd is open and X is continuous from the right (càd) and adapted then
T := inf{t > 0 : X t ∈ Λ} is a stopping time.
(iii) If Λ ⊆ Rd is closed and X is càd and adapted then T := inf{t > 0 : X t ∈
Λ or X t− ∈ Λ} is a stopping time.
(iv) If X is càdlàg and adapted and ∆X T 1 T <∞ = 0 for all stopping times T then
∆X is indistinguishable from the zero process.
PROOF: Read the proofs of these facts as an exercise.
ƒ
6
Preliminaries
1.1.9 Definition. O = σ(X : X is adapted and càdlàg) is the optional σ-algebra.
A stochastic process X is an optional process if X is O -measurable.
1.1.10 Theorem (Début theorem). If A ∈ O then T (ω) := inf{t : (ω, t) ∈ A}, the
début time of A, is a stopping time.
Remark. This theorem requires that the filtration is right continuous. For example,
suppose that T is the hitting time of an open set by a left continuous process. Then
you can prove {T < t} ∈ F t for all t without using right continuity of the filtration,
but you cannot necessarily prove that {T = t} ∈ F t without it. You need to “look
into the future” a little bit.
1.1.11 Corollary. If X is optional and B ⊆ Rd is a Borel set then the hitting time
T := inf{t > 0 : X t ∈ B} is a stopping time.
1.1.12 Theorem. If X is an optional process then
(i) X is (F ⊗ B([0, ∞)))-measurable and
(ii) X T 1 T <∞ ∈ F T for any stopping time T .
In particular, (X t∧T ) t≥0 is also an optional process, i.e. optional processes are “stable under stopping”.
1.1.13 Theorem (Monotone class theorem).
Suppose that H is collection of bounded R-valued functions such that
(i) H is a vector space.
(ii) 1Ω ∈ H, i.e. constant functions are in H.
(iii) If ( f n )n≥0 ⊆ H is monotone increasing and f := limn→∞ f n (pointwise) is
bounded then f ∈ H.
(In this case H is called a monotone vector space.) Let M be a multiplicative collection of bounded functions (i.e. if f , g ∈ M then f g ∈ M ). If M ⊆ H then H
contains all the bounded functions that are measurable with respect to σ(M ).
PROOF (OF THEOREM 1.1.12): Use the Monotone Class Theorem. Define
M := {X : Ω × [0, ∞) → R | X is càdlàg, adapted, and bounded}
H := {X : Ω × [0, ∞) → R | X is bounded and (i) and (ii) hold}
It can be checked that H is a monotone vector space, M is a multiplicative collection, and σ(M ) = O . If we prove that M ⊆ H then we are done. Let X ∈ M and
define
∞
X
X (n) :=
X kn 1[ k−1n , kn ) .
k=1
2
2
2
Since X is right continuous, X (n) → X pointwise. Let B be a Borel set.
{X (n) ∈ B} =
∞ n
[
k=1
X
k
2n
o k −1 k ∈B ×
,
∈ F ⊗ B([0, ∞))
2n 2n
1.2. Review of martingales
7
This proves that X satisfies (i). To prove (ii), let T be a stopping time and define
¨k
if k−1
≤ T < 2kn
n
2n
Tn := 2
∞ if T = ∞.
Then the Tn ’s are stopping times and Tn ↓ T .
∞ n
[
{X Tn ∈ B} ∩ {Tn ≤ t} =
X
o
k
2n
∈ B ∩ Tn =
k=1
k
≤t
2n
k
2n
∈ Ft
This shows that X Tn ∈ F Tn . Since X is right continuous,
X T 1 T <∞ = lim X Tn 1 Tn <∞ .
n→∞
It can be shown that, since the filtration is right continuous,
X T 1 T <∞ ∈ F T and X satisfies (ii). Therefore X ∈ H.
T∞
n=1 F Tn
= F T , so
ƒ
If X is càdlàg and adapted then an argument similar to that in the previous
proof shows that X |Ω×[0,t] ∈ F t ⊗ B([0, t]) for all t, i.e. X is progressively measurable. By a similar monotone class argument, it can be shown that every optional
process is a progressive process.
1.1.14 Definition. V := σ(X : X is progressively measurable) is the progressive
σ-algebra.
1.1.15 Corollary. O ⊆ V .
1.2
Review of martingales
1.2.1 Theorem. Let X be a (sub-, super-) martingale and assume that F satisfies
the usual conditions. Then X has a right continuous modification if and only if the
function t 7→ E[X t ] is right continuous. Furthermore, this modification has left
limits everywhere.
PROOF (SKETCH): The process Xet := lim s↓t X s is the correct modification.
s∈Q
ƒ
1.2.2 Corollary. Every martingale has a càdlàg modification, unique up to indistinguishability.
1.2.3 Theorem. Let X be a right continuous sub-martingale with sup t≥0 E[X t+ ] <
∞. Then X ∞ := lim t→∞ X t exists a.s. and X ∞ ∈ L 1 .
1.2.4 Definition. A collection of random variables (Uα )α∈A is uniformly integrable
or u.i. if
lim sup E[1|Uα |>n Uα ] = 0.
n→∞ α∈A
8
Preliminaries
1.2.5 Theorem. The following are equivalent for a family (Uα )α∈A.
(i) (Uα )α∈A is u.i.
(ii) supα∈A E[|Uα |] < ∞ and for all " > 0 there is δ > 0 such that if P[Λ] < δ
then supα∈A E[1Λ Uα ] < ".
(iii) (de la Vallée-Poussin criterion) There is a positive, increasing, convex funcG(x)
tion G on [0, ∞) such that lim x→∞ x = ∞ and supα∈A E[G(|Uα |)] < ∞.
1.2.6 Theorem. The following are equivalent for a martingale X .
(i) (X t ) t≥0 is u.i.
(ii) X is closable, i.e. there is an integrable r.v. Z such that X t = E[Z|F t ].
(iii) X converges in L 1 .
(iv) X ∞ = lim t→∞ X t a.s. and in L 1 and X ∞ closes X .
Remark.
(i) If X is u.i. then X is bounded in L 1 (but not vice versa), so Theorem 1.2.3
implies that X t → X ∞ a.s. Theorem 1.2.6 upgrades this to convergence in
L 1 , i.e. E[|X t − X ∞ |] → 0.
W
(ii) If X is closed by Z then E[Z|F∞ ] also closes X , where F∞ := t≥0 F t .
Furthermore, X ∞ = E[Z|F∞ ].
1.2.7 Example (Simple random walk). Let (Zn )n≥1 be i.i.d. with
P[Zn = 1] = P[Zn = −1] =
Take F t := σ{Zk : k ≤ t} and X t :=
Pbtc
k=1
1
2
.
Zk . Then X is not a closable martingale.
1.2.8 Example. Let M t := exp(B t − 12 t), the stochastic exponential of Brownian
motion, a martingale. Then M t → 0 a.s. by Theorem 1.2.3 because it is a positive
valued martingale (hence −M is a sub-martingale with no positive part). But
E[|M t |] = 1 for all t so M is not u.i. by Theorem 1.2.6.
1.2.9 Theorem.
(i) If X is a closable (sub-, super-) martingale and S ≤ T are stopping times
then E[X T |FS ] = X S (≥, ≤).
(ii) If X is a (sub-, super-) martingale and S ≤ T are bounded stopping times
then E[X T |FS ] = X S (≥, ≤).
(iii) If X is a right continuous sub-martingale and p > 1 then
p
sup kX t k L p .
k sup X t k L p ≤
p − 1 t≥0
t≥0
In particular if p = 2 then E[sup t≥0 X t2 ] ≤ 4 sup t≥0 E[X t2 ].
(iv) (Jensen’s inequality) If ϕ is a convex function, Z is an integrable r.v., and G
is a sub-σ-algebra of F then ϕ(E[Z|G ]) ≤ E[ϕ(Z)|G ].
1.3. Poisson process and Brownian motion
1.2.10 Definition. Let X be a process and T be a random time. The stopped
process is X tT := X t 1 t≤T + X T 1 t>T,T <∞ = X T ∧t .
If T is a stopping time and X is càdlàg and adapted then so is X T .
1.2.11 Theorem.
(i) If X is a u.i. martingale and T is a stopping time then X T is a u.i. martingale.
(ii) If X is a martingale and T is a stopping time then X T is a martingale.
That is, martingales are “stable under stopping”.
1.2.12 Definition. Let X be a martingale.
(i) If X t ∈ L 2 for all t then X is called a square integrable martingale.
(ii) If (X t ) t∈[0,∞) is u.i. and X ∞ ∈ L 2 then X is called an L 2 -martingale.
1.2.13 Exercise. Any L 2 -martingale is a square integrable martingale, but not
conversely.
SOLUTION: Let X be an L 2 -martingale. Then X ∞ ∈ L 2 and so, by the conditional
version of Jensen’s inequality,
2
2
E[X t2 ] = E[(E[X ∞ |F t ])2 ] ≤ E[E[X ∞
|F t ]] = E[X ∞
] < ∞.
Therefore X t ∈ L 2 for all t. We have already seen that the stochastic exponential
of Brownian motion is not u.i. (and hence not an L 2 -martingale) but it is square
integrable because the normal distribution has a finite valued moment generating
function.
ú
1.3
Poisson process and Brownian motion
1.3.1 Definition. Suppose that (Tn )n≥1 is a strictly
increasing sequence of ranP
dom times with T1 > 0 a.s. The process Nt = n≥1 1 Tn ≤t is the counting process
associated with (Tn )n≥1 . The random time T := supn≥1 Tn is the explosion time. If
T = ∞ a.s. then N is a counting process without explosion.
1.3.2 Theorem. A counting process is an adapted process if and only if Tn is a
stopping time for all n.
PROOF: If N is adapted then {t < Tn } = {Nt < n} ∈ F t for all t and all n. Therefore, for all n, {Tn ≤ t} ∈ F t for all t, so Tn is a stopping time. Conversely, if all
the Tn are stopping times then, for all t, {Nt ≤ n} = {t ≤ Tn } ∈ F t for all n. Since
N takes only integer values this implies that Nt ∈ F t .
ƒ
1.3.3 Definition. An adapted process N is called a Poisson process if
() N is a counting process.
(i) Nt −Ns is independent of Fs for all 0 ≤ s < t < ∞ (independent increments).
9
10
Preliminaries
(d)
(ii) Nt − Ns = Nt−s for all 0 ≤ s < t < ∞ (stationary increments).
Remark. Implicit in this definition is that a Poisson process does not explode. The
definition can be modified slightly to allow this possibility, and then it can be
proved as a theorem that a Poisson process does not explode, but the details are
very technical and relatively unenlightening.
1.3.4 Theorem. Suppose that N is a Poisson process.
(i) N is continuous in probability.
(d)
(ii) Nt = Poisson(λt) for some λ ≥ 0, called the intensity or arrival rate of N . In
particular, Nt has finite moments of all orders for all t.
(iii) (Nt − λt) t≥0 and ((Nt − λt)2 − λt) t≥0 are martingales.
(iv) If F tN := σ(Ns : s ≤ t) and FN = (F tN ) t≥0 then FN is right continuous.
PROOF: Let α(t) := P[Nt = 0] for t ≥ 0. For all s < t,
α(t + s) = P[Nt+s = 0]
= P[{Ns = 0} ∩ {Nt+s − Ns = 0}]
non-decreasing and non-negative
= P[Ns = 0] P[Nt+s − Ns = 0]
independent increments
= P[Ns = 0] P[Nt = 0]
stationary increments
= α(t)α(s)
If t n ↓ t then {Nt n = 0} % {Nt = 0}, so α is right continuous and decreasing. It
follows that either α ≡ 0 or α(t) = e−λt for some λ ≥ 0. By the definition of
counting process N0 = 0, so α is cannot be the zero function.
(i) Observe that, given " > 0 small, for all s < t,
P[|Nt − Ns | > "] = P[|Nt−s | > "]
stationary increments
= P[Nt−s > "]
N is non-decreasing
= 1 − P[Nt−s = 0]
N is integer valued
=1−e
−λ(t−s)
→ 0 as s → t.
Therefore N is left continuous in probability. The proof of continuity from
the right is similar.
(ii) First we need to prove that lim t→0 1t P[Nt = 1] = λ. Towards this, let
β(t) := P[Nt ≥ 2] for t ≥ 0. If we can show that lim t→0 β(t)/t = 0 then we
would have
P[Nt = 1]
1 − α(t) − β(t)
lim
= lim
= λ.
t→0
t→0
t
t
It is enough to prove that limn→∞ nβ(1/n) = 0. Divide [0, 1] into n equal
subintervals and let Sn be the number of subintervals with at least two ar(d)
rivals. It can be see that Sn = Binomial(n, β(1/n)) because of the stationary
and independent increments of N . In the definition of counting process the
1.3. Poisson process and Brownian motion
11
sequence of jump times is strictly increasing, so limn→∞ Sn = 0 a.s. Clearly
Sn < N1 , so limn→∞ nβ(1/n) = limn→∞ E[Sn ] = 0 provided that E[N1 ] < ∞.
We are going to gloss over this point.
Let ϕ(t) := E[γNt ] for 0 < γ < 1.
ϕ(t + s) = E[γNt+s ] = E[γNs γNt+s −Ns ] = E[γNs ] E[γNt ] = ϕ(t)ϕ(s)
and ϕ is right continuous because N has right continuous paths. Therefore
ϕ(t) = e tψ(γ) for some function ψ of γ. By definition,
ϕ(t) =
X
γn P[Nt = n]
n≥0
e
tψ(γ)
= α(t) + γ P[Nt = 1] +
X
γn P[Nt = n].
n≥2
Differentiating, ψ(γ) = lim t→0 (ϕ(t) − 1)/t = −λ + γλ by the computations
above. Comparing coefficients of γn in the power series shows that Nt has a
Poisson distribution with rate λt, i.e. P[Nt = n] = e−λt (λt)n /n!.
(iii) Exercise.
(iv) See the textbook.
ƒ
1.3.5 Definition. An adapted process B is called a Brownian motion if
(i) B t −Bs is independent of Fs for all 0 ≤ s < t < ∞ (independent increments).
(d)
(ii) B t − Bs = Normal(0, t − s) for all 0 ≤ s < t < ∞ (stationary, normally
distributed, increments).
If B0 ≡ 0 then B is a standard Brownian motion.
1.3.6 Theorem. Let B be a Brownian motion.
(i) If E[|B0 |] < ∞ then B is a martingale.
(ii) There is a modification of B with continuous paths.
(iii) (B 2t − t) t≥0 is a martingale when B is standard BM.
(iv) Let Πn be a refining sequence of partitions
of the interval [a, a + t] with
P
limn→∞ mesh(Πn ) = 0. Then Πn B := t n ∈Πn (B t i+1 − B t i )2 → t in L 2 and a.s.
when B is standard BM.
(v) For almost all ω, the function t 7→ B t (ω) is of unbounded variation.
P
Remark. To prove (ii) with the additional assumption that n≥0 mesh(Πn ) < ∞,
but dropping the assumption that the partitions are refining, you can use the BorelCantelli lemma. The proof of (iv) uses the backwards martingale convergence
theorem.
12
Preliminaries
1.4
Lévy processes
1.4.1 Definition. An adapted, real-valued process X is called a Lévy process if
() X 0 = 0
(i) X t −X s is independent of Fs for all 0 ≤ s < t < ∞ (independent increments).
(d)
(ii) X t − X s = X t−s for all 0 ≤ s < t < ∞ (stationary increments).
(iii) (X t ) t≥0 is continuous in probability.
If only (i) and (iii) hold then X is an additive process. If (X t ) t≥0 is non-decreasing
then X is called a subordinator.
Remark. If the filtration is not specified then we assume that F = FX is the natural
(not necessarily complete) filtration of X . In this case X might then be called an
intrinsic Lévy process.
1.4.2 Example. The Poisson process and Brownian motion are both Lévy processes. Let W be a standard BM and define Tb := inf{t > 0 : Wt ≥ b}. Then
(Tb ) b≥0 is an intrinsic Lévy process and subordinator for the filtration (F Tb ) b≥0 .
Indeed, if 0 ≤ a < b < ∞ then Ta − Tb is independent of F Ta and distributed as
Tb−a by the strong Markov property of W . We will see that T is a stable process
with parameter α = 1/2.
1.4.3 Theorem. Let X be a Lévy process. Then f t (z) := E[e izX t ] = e−tΨ(z) for
some continuous function Ψ = ΨX . Furthermore, M tz := e izX t / f t (z) is a martingale
for all z ∈ R.
PROOF: Fix z ∈ R. By the stationarity and independence of the increments,
f t (z) = f t−s (z) fs (z) for all 0 ≤ s < t < ∞.
(1.1)
We would like to show that t 7→ f t (z) is right-continuous. By the multiplicative
property (1.1), it suffices to show t 7→ f t (z) is right continuous at zero. Suppose
that t n ↓ 0; we need to show that f t n (z) → 1. By definition, | f t n (z)| ≤ 1, so we are
done if we can show that every convergent subsequence converges to 1. Suppose
without loss of generality that f t n (z) → a ∈ C. Since X is continuous in probability,
e izX t n → 1 in probability. Along a subsequence we have convergence almost surely,
i.e. f t t n (z) → 1. Therefore a = 1 and we are done.
k
The multiplicative property and right continuity imply that f t (z) = e−tΨ(z) for
some number Ψ(z). Since f t (z) is a characteristic function, z 7→ f t (z) is continuous (use dominated convergence). Therefore Ψ must be continuous as well. In
particular, f t (z) 6= 0 for all t and all z. Let 0 ≤ s < t < ∞.
izX t e
e izX s
F =
E[e iz(X t −X s ) |Fs ]
independent increments
E
s
f t (z) f t (z)
=
e izX s
f t (z)
f t−s (z)
stationary increments
1.4. Lévy processes
=
13
e izX s
fs (z)
f is multiplicative.
Therefore M tz is a martingale.
ƒ
1.4.4 Theorem. If X is an additive process then X is Markov with transition function Ps,t (x, B) = P[X t − X s ∈ B − x]. In particular, X is spatially homogeneous, i.e.
Ps,t (x, B) = Ps,t (0, B − x).
1.4.5 Corollary. If X is additive, " > 0, and T < ∞ then limu↓0 α",T (u) = 0, where
α",T is defined in the next theorem.
PROOF: Ps,t (x, B" (x)C ) = P(|X t − X s | ≥ ") → 0 as s → t uniformly on [0, T ] in
probability. It is an exercise to show that the continuity in probability of X implies
uniform continuity in probability on compact intervals.
ƒ
1.4.6 Theorem. Let X be a Markov process on Rd with transition function Ps,t . If
limu↓0 α",T (u) = 0, where
α",T (u) = sup{Ps,t (x, B" (x)C ) : x ∈ Rd , s, t ∈ [0, T ], 0 ≤ t − s ≤ u},
then X has a càdlàg modification. If furthermore limu↓0 α",T (u)/u = 0 then X has
a continuous modification.
PROOF: See Lévy processes and infinitely divisible distributions by Kin-Iti Sato. The
important steps are as follows.
Fix " > 0 and ω ∈ Ω. Say that X (ω) has an "-oscillation n-times in M ⊆ [0, ∞)
if there are t 0 < t 1 < · · · < t n all in M such that |X t j (ω) − X t j−1 (ω)| ≥ ". X has
"-oscillation infinitely often in M if this holds for all n. Define
Ω02 :=
∞
∞ \
\
N =1 k=1
{ω : X (ω) does not have 1k -oscillation infinitely often in [0, N ] ∩ Q}
It can be shown that Ω02 ∈ F , and also that
n
o
Ω02 ⊆ ω : lim X s exists for all t and lim X s exists for all t > 0 .
s↓t,s∈Q
The hard part is to show that P[Ω02 ] = 1.
s↑t,s∈Q
ƒ
Remark.
(i) If X is a Feller process then X has a càdlàg modification (see Revuz-Yor).
(ii) From now on we assume that we are working with a càdlàg version of any
Lévy process that appears.
(iii) P[X t 6= X t− ] = 0 for any process X that is continuous in probability. Of
course, this does not mean that X doesn’t jump, it means that X has no fixed
times of discontinuity.
14
Preliminaries
1.4.7 Theorem. Let X be a Lévy process and N be the collection of P-null sets. If
G t := F tX ∨ N for all t then G = (G t ) t≥0 is right continuous.
PROOF: Let (s1 , . . . , sn ) and (u1 , . . . , un ) be arbitrary vectors in (R+ )n and Rn respectively. We first want to show that
E[e i(u1 X s1 +···+un X sn ) |G t ] = E[e i(u1 X s1 +···+un X sn ) |G t+ ].
It is clear that it suffices to show this with s1 , . . . , sn > t. We take n = 2 for
notational simplicity, and assume that s2 ≥ s1 > t.
E[e i(u1 X s1 +u2 X s2 ) |G t+ ] = lim E[e i(u1 X s1 +u2 X s2 ) |Gw ]
w↓t
exercise
= lim E[e iu1 X s1 Msu22 |Gw ] fs2 (u2 )
w↓t
= lim E[e iu1 X s1 Msu12 |Gw ] fs2 (u2 )
tower property
= lim E[e i(u1 +u2 )X s1 |Gw ] fs2 −s1 (u2 )
f is multiplicative
w↓t
w↓t
= lim
w↓t
e i(u1 +u2 )X w
f w (u1 + u2 )
fs1 (u1 + u2 ) fs2 −s1 (u2 )
= e i(u1 +u2 )X t fs1 −t (u1 + u2 ) fs2 −s1 (u2 )
M is a martingale
X is càdlàg
By the same steps, we obtain
E[e i(u1 X s1 +u2 X s2 ) |G t ] = e i(u1 +u2 )X t fs1 −t (u1 + u2 ) fs2 −s1 (u2 ).
X
By a monotone class argument, E[Z|G t ] = E[Z|G t+ ] for any bounded Z ∈ F∞
. It
follows that G t+ \ G t consists only of sets from N . Since N ⊆ G t , they must be
equal.
ƒ
1.4.8 Corollary (Blumenthal 0-1 Law). Let X be a Lévy process. If A ∈
then P[A] = 0 or 1.
T
X
t>0 F t
1.4.9 Theorem. If X is a Lévy process and T is a stopping time then, on {T < ∞},
Yt := X T +t − X T is a Lévy process with respect to (F t+T ) t≥0 and Y has the same
finite dimensional distributions as X .
1.4.10 Corollary. If X is a Lévy process then X is a strong Markov process.
1.4.11 Theorem. If X is a Lévy process with bounded jumps then E[|X t |n ] < ∞
for all t and all n.
PROOF: Suppose that sup t |∆X t | ≤ C. Define a sequence of random times Tn
recursively as follows. T0 ≡ 0 and
¨
inf{t > Tn : |X t − X Tn | ≥ C} if Tn < ∞
Tn+1 =
∞
if Tn = ∞
1.5. Lévy measures
15
By definition of the Tn , sups |X sTn | ≤ 2nC. The 2 comes from the possibility that X
could jump by C just before hitting the stopping level. Since X is càdlàg, the Tn
are all stopping times. Further, since X is a strong Markov process on {Tn < ∞},
(i) Tn − Tn−1 is independent of F Tn−1 and
(d)
(ii) Tn − Tn−1 = T1 .
Therefore
E[e−Tn ] = E
Y
n
e−(Tk −Tk−1 ) ≤ (E[e−T1 ])n =: αn ,
k=0
where 0 ≤ α < 1. The ≤ comes from the fact that e−∞ = 0 and some of the Tn may
be ∞. (We interpret Tk − Tk−1 to be ∞ if both of them are ∞.) By Chebyshev’s
inequality,
E[e−Tn ]
≤ αn e t .
P[|X t | > 2nC] ≤ P[Tn < t] ≤
e−t
Finally,
E[e
β|X t |
]≤1+
≤1+
∞
X
n=0
∞
X
E[eβ|X t | 12nC<|X t |≤2(n+1)C ]
e2β(n+1)C ) P[|X t | > 2nC]
n=0
≤ 1 + e2β C e t
∞
X
(αe2β C )n
n=0
Choosing an appropriately small, positive β shows that |X t | has an exponential
moment, so it has polynomial moments of all orders.
ƒ
1.5
Lévy measures
Suppose that Λ ∈ B(R) and 0 ∈
/ Λ. Let X be a Lévy process (with càdlàg paths, as
always) and define inductively a sequence of random times TΛ0 ≡ 0 and
TΛn+1 := inf{t > TΛn : ∆X t ∈ Λ}.
These times have the following properties.
(i) {TΛn ≥ t} ∈ F t+ = F t by the usual conditions and (TΛn )n≥1 is an increasing
sequence of (possibly ∞-valued) stopping times.
(ii) TΛ1 > 0 a.s. since X 0 ≡ 0, X has càdlàg paths, and 0 ∈
/ Λ.
n
(iii) limn→∞ TΛ = ∞ since X has càdlàg paths and 0 ∈
/ Λ (see Homework 1,
problem 10).
Let N Λ be the number of jumps with size in Λ before time t.
NtΛ
:=
X
0<s≤t
1Λ (∆X s ) =
∞
X
n=1
1 TΛn ≥t
16
Preliminaries
Since X is a strong Markov process, N Λ has stationary and independent increments. It is also a counting process with no explosions, so it is a Poisson process.
By Theorem 1.1.12(i) we can define ν(Λ) := E[N1Λ ], the intensity of N Λ . We can
/ Λ by taking ν(Λ) = ∞
extend thisP
definition to the case when 0 ∈ Λ, so long as 0 ∈
whenever 0<s≤t 1Λ (∆X s ) would fail to converge. Define ν({0}) = 0.
1.5.1 Theorem. For each t and ω the map Λ 7→ NtΛ (ω) is a Borel counting measure on R \ {0}, and ν(Λ) := E[N1Λ ] is also a Borel measure on R \ {0}.
1.5.2 Definition. ν is the Lévy measure associated with X .
1.5.3 Example. If X is a Poisson process of rate λ then ν = λδ1 .
1.5.4 Theorem. If X is a Lévy process then it can be decomposed as X =
TY + Z
where Y is a Lévy process and martingale with bounded jumps (so Yt ∈ p≥1 L p
for all t) and Z is a Lévy process with paths of finite variation on compact subsets
of [0, ∞).
1.5.5 Theorem. Let X be a Lévy process with jumps bounded by C. The process
Z t := X t − E[X t ] is a martingale that can be decomposed as Z = Z c + Z d , where
Z c and Z d are independent Lévy processes. Moreover,
Z
x(N (t, d x) − tν(d x)),
Z td :=
|x|≤C
where N (t, d x) is the measure of Theorem 1.5.1, and the remainder Z c has continuous paths.
1.5.6 Lemma. If X is a subordinator with continuous paths then X t = c t for some
constant c.
PROOF: X 0 (0) = lim"↓0 X " /" exists a.s., and since X 0 (0) ∈ F0 , it is a constant a.s.
Let δ > 0 and inductively define a sequence of stopping times by T0 = 0 and
Tk+1 := inf{t > Tk : X t − X Tk ≥ (c + δ)t}.
Then, since X is continuous, X T1 = (c + δ)T1 and X t ≤ (c + δ)t for t ≤ T1 .
Additionally, X t ≤ (c + δ)t for t ≤ Tk . Because X is a Lévy process, (Tk+1 − Tk )k≥0
are strictly positive i.i.d. random variables. By the strong law of large numbers
Tk → ∞ as k → ∞, so X t ≤ (c + δ)t for all t. Therefore X t ≤ c t for all t, and the
proof that X t ≥ c t for all t is similar.
ƒ
1.5.7 Theorem (Lévy-Itô decomposition). If X is a Lévy process then there are
constants σ and α such that
Z
X
(d)
x(N (t, d x) − tν(d x)) +
∆X s 1|∆X s |≥1
X t = σB t + αt +
|x|<1
0<s≤t
where B is a standard Brownian motion and α = E[X 1 −
R
|x|≥1
x N (1, d x)].
1.5. Lévy measures
17
Remark. X has finite variation on compacts if either
(i) σ = 0 and ν(R) < ∞ or R
(ii) σ = 0 and ν(R) = ∞ but |x|<1 |x|ν(d x) < ∞.
R
If σ 6= 0 or |x|<1 |x|ν(d x) = ∞ then X has infinite variation on compacts. To
prove that the continuous part of X is σB we will need Lévy’s characterization of
BM, proven later.
1.5.8 Lemma. Let X be a Lévy process, Λ ⊆ R \ {0} be Borel with 0 ∈
/ Λ, and let
f be Borel measurable and finite valued on Λ.
(i) The process Y defined by
Z
X
Yt :=
f (x)N (t, d x) =
f (∆X s )1∆X s ∈Λ
Λ
0<s≤t
is a Lévy process. (Show as an exercise that Y is continuous in probability.)
(ii) Let M t := X t − J tΛ where J Λ is defined by
Z
X
f (∆X s )1∆X s ∈Λ .
x N (t, d x) =
J tΛ :=
Λ
0<s≤t
Then M is a Lévy process with jumps outside of Λ. In particular, if we take
Λ = {x : |x| ≥ 1} then M
R is a Lévy process with
R bounded jumps.
(iii) If f 1Λ ∈ L 1 (ν) then E[ Λ f (x)N (t, d x)] = t Λ f (x)ν(d x)]. If f 1Λ ∈ L 2 (ν)
R
R
R
then E[( Λ f (x)N (t, d x) − t Λ fP
(x)ν(d x))2 ] = t Λ f (x)2 ν(d x).
(iv) If Λ1 ∩ Λ2 = ∅
Pthen the process ( 0<s≤t ∆X s 1Λ1 (∆X s )) t≥0 is independent of
the process ( 0<s≤t ∆X s 1Λ2 (∆X s )) t≥0 .
P
PROOF (OF THEOREM 1.5.4): Let J t := 0<s≤t ∆X s 1|∆X s |≥1 . J has finite variation
on compacts since it is increasing and jumps finitely many times in any bounded
interval. By lemma (ii), X − J is a Lévy process with bounded jumps. It can
be shown (as an exercise) that E[X t − J t ] = αt, where α := E[X 1 − J1 ]. Then
Yt := X t − J t − αt is a martingale and Z t := J t + αt is a Lévy process with paths of
finite variation on compacts.
X t = X t − J t − αt + J t + αt .
{z
} | {z }
|
Yt
Zt
ƒ
PROOF (OF THEOREM 1.5.5): That Z t := X t − E[X t ] is a martingale is an exercise.
Suppose without loss of generalityRthat the jumps ofR X are bounded by 1. Define
Λ
1
Λk := { k+1
< |x| ≤ 1k } and M t k := Λ x N (t, d x)−t Λ xν(d x). Then the M Λk are
k
k
martingales by lemma (ii) P
and are in L 2 by lemma (iii) and they are independent
n
n
by lemma (iv). Let M := k=1 M Λk .
Z
Z
n
n
X
X
n
Λk
2
x 2 ν(d x)
Var(M ) =
Var(M ) =
t
x ν(d x) = t
k=1
k=1
Λk
1
<|x|≤1
n+1
18
Preliminaries
Prove as an exercise that Z − M n is independent of M n (it is not an immediate
consequence of the lemma). Therefore Var(M n ) = Var(Z t ) − Var(Z − M n ) < ∞
since Var(Z t ) < ∞ because Z has bounded jumps.
Since M n is a sum of independent random variables we can take the limit as
n → ∞ in L 2 (fill in the details as an exercise). Let Z c := limn→∞ (Z − M n ) and
Z d := limn→∞ M n . Then Z c is independent of Z d since Z − M n is independent of
M n for all n. The formula for Z d is immediate.
1
Something is missing from the last
.
To show Z c is continuous, note first that Z − M n has jumps bounded by n+1
part of this proof.
n
2
By Doob’s maximal inequality sup0<s≤t (Zs − Ms ) converges in L . Along a subsequence Z − M n → Z c uniformly on compacts so Z c is continuous.
ƒ
1.5.9 Theorem (Lévy-Khintchine formula). Let X be a Lévy process with characteristic function E[e izX t ] = e−tΨ(z) . Then the Lévy-Khintchine exponent is
Z
σ2 z 2
− iαz + (1 − e iz x + iz x1|x|<1 )ν(d x).
Ψ(z) =
2
R
PROOF (SKETCH): In the Lévy-Itô decomposition it can be shown that the parts are
all independent, so the characteristic functions multiply. Show as an exercise that
the characteristic function of
R
X
t
(e iz x −1)ν(d x)
∆X s 1|∆X s |≥1 is e |x|≥1
0<s≤t
and the characteristic function of
Z
x(N (t, d x) − tν(d x))
|x|<1
is
e
t
R
|x|<1
(1−e iz x +iz x)ν(d x)
.
ƒ
1.5.10 Definition. A probability distribution F on R is an infinitely divisible distri(d)
bution if for all n there are X 1 , . . . , X n i.i.d. such that X 1 + · · · + X n = F .
1.5.11 Theorem.
(i) If X is a Lévy process then, for all t, X t has an infinitely divisible distribution.
(ii) Conversely, if F is an infinitely divisible distribution then there is a Lévy
(d)
process X such that X 1 = F .
(iii) If F is an infinitely divisible distribution then its characteristic function has
the form of the Lévy-KhintchineRexponent with t = 1, where ν is a measure
on R such that ν({0}) = 0 and R (|x|2 ∧ 1)ν(d x) < ∞, and the representation of F in terms of (σ, α, ν) is unique.
1.5.12 Examples.
iz
(i) If X ∼ Poisson(λ) then the characteristic function is eλt(e −1) , so σ = α = 0
and ν = λδ1 .
1.5. Lévy measures
19
(ii) The Γ-process is the Lévy process such that X 1 has a Γ-distribution, where
bc
P[X 1 ∈ d x] = Γ(c)
x c−1 e bx 1 x>0 d x for some constants b, c > 0. The LévyKhintchine exponent is
Z
(e iz x − 1)
c
e−bx
R
x
1 x>0 d x.
In this case ν(d x) = e−bx /x1 x>0 d x, σ = 0, and α = bc (1 − e−b ) > 0. The
paths of X are non-decreasing since it has positive drift, no volatility, and
positive jumps. Because ν((0, 1)) = ∞, X has infinite activity.
(iii) Stable processes are those with Lévy-Khintchine exponent of the form
izδ + m1
Z
∞
e
iz x
−1−
0
iz x
1+
1
x2
dx
x 1+γ
+ m2
Z
0
e
iz x
−1−
−∞
iz x
1+
x2
1
x 1+γ
dx
where 0 < γ < 2. When m1 = m2 one can take ν(d x) = |x|11+γ d x. In this
case there are Y1 , Y2 , . . . i.i.d. and constants an , and bn such that
∞
1 X
an
(w)
Yi − bn −→ X 1
i=1
R
(d)
−1
and (β γ X β t ) t≥0 = (X t ) t≥0 for all β > 0. If 1 ≤ α < 2 then |x|ν(d x) = ∞
so the process would have infinite variation on compacts.
(iv) For the hitting time process (Tb ) b≥0 of standard Brownian motion B, we
have
b
b2
P[Tb ∈ d t] = p
e− 2t 1 t>0 d t
2πt 3
so the Lévy-Khintchine exponent is
b
p
2π
Z
∞
3
(e iz x − 1)x − 2 d x.
0
It follows that T is a stable process with γ = 12 . Indeed,
1
β2
Tβ b =
1
β2
inf{t : B t = β b}
= inf{t/β 2 : B t /β = b}
(d)
= inf{t : Bβ 2 t /β = b} = Tb
(d)
since (Bβ 2 t /β) t≥0 = B. Also note that T is non-decreasing.
20
Preliminaries
1.6
Localization
1.6.1 Definition.
(i) If C is a collection of processes then the localized class, Cloc , is the collection
of processes X such that there exists a sequence (Tn )n≥1 of stopping times
such that Tn ↑ ∞ a.s. and X Tn ∈ C for all n. The sequence (Tn )n≥1 is called a
localizing sequence for X relative to C.
(ii) If C is the collection of adapted, càdlàg, uniformly integrable martingales
then an element of the localized class is called a local martingale.
(iii) A stopping time T reduces X relative to C if X T ∈ C.
(iv) C is stable under stopping if X T ∈ C for all X ∈ C and all stopping times T .
1.6.2 Theorem. Suppose that C is a collection of processes that is stable under
stopping.
(i) Cloc is stable under stopping and (Cloc )loc = Cloc . In particular, a local local
martingale is a local martingale.
(ii) If T reduces X relative to C and S ≤ T a.s. then S reduces X .
(iii) If M and N are local martingales then M + N is a local martingale. If S and
T reduce M then S ∨ T reduces M .
(iv) (C ∩ C 0 )loc = Cloc ∩ C 0 loc .
PROOF (OF (I)): Let X ∈ Cloc and T be a stopping time. If (Tn )n≥1 is a localizing
sequence for X then (X T ) Tn = (X Tn ) T ∈ C since C is stable under stopping. Therefore (Tn )n≥1 is a localizing sequence for X T and it is seen that Cloc is stable under
stopping.
Now let X ∈ (Cloc )loc and (Tn ) be a localizing sequence for X relative to Cloc ,
so that X Tn ∈ Cloc for all n. Then for each n there are (Tn,p ) p≥1 such that Tn,p ↑ ∞
a.s. as p → ∞ and (X Tn ) Tn,p ∈ C for all p. For all n there is pn such that
P[Tn,pn < Tn ∧ n] ≤
1
2n
since a.s. convergence implies
Vconvergence in probability. Define a new sequence
of stopping times Sn := Tn ∧ m≥n Tm,pm . Then, for all n, Sn ≤ Sn+1 and
P[Sn < Tn ∧ n] ≤
∞
X
1
m=n
2m
=
1
2n−1
.
By the Borel-Cantelli lemma, P[Sn < Tn ∧ n i.o.] = 0, which implies that Sn ↑ ∞
a.s. Finally, X Sn = ((X Tn ) Tn,pn )Sn ∈ C since C is stable under stopping, so (Sn )n≥1 is
a localizing sequence for X relative to C.
ƒ
1.6.3 Corollary.
(i) If X is a locally square integrable martingale then X is a local martingale
(but the converse does not hold).
(ii) If M is a process and there are stopping times (Tn )n≥1 such that Tn ↑ ∞ a.s.
and M Tn is a martingale for all n then M is a local martingale.
1.7. Integration with respect to processes of finite variation
We will see that it is important to be able to decide when a local martingale a
“true” martingale. To this end, let X t∗ := sups≤t |X s | and X ∗ = sups≥0 |X s |.
1.6.4 Theorem. Let X be a local martingale.
(i) If E[X t∗ ] < ∞ for all t then X is a martingale.
(ii) If E[X ∗ ] < ∞ then X is a u.i. martingale.
PROOF: Let (Tn )n≥1 be a localizing sequence for X . For any s ≤ t,
E[X Tn ∧t |Fs ] = X Tn ∧t
by the optional stopping theorem. Since X Tn ∧t is dominated by the integrable random variable X t∗ , we may apply the (conditional) dominated convergence theorem
to both sides. Therefore X is a martingale. If E[X ∗ ] < ∞ then the family (X t ) t≥0
is dominated by the integrable random variable X ∗ , so it is u.i.
ƒ
1.6.5 Corollary.
(i) If X is a bounded local martingale then it is a u.i. martingale.
(ii) If X is a local martingale and a Lévy process then X is a martingale.
(iii) If (X n )n≥1 is a discrete time local martingale and E |X n | < ∞ for all n then X
is a martingale.
1.6.6 Example. Suppose (An )n≥1 is a measurable partition of Ω with P[An ] = 2−n
for all n. Further suppose (Zn )n≥1 is a sequence of random variables independent
of the An and such that P[Zn = 2n ] = P[Zn = −2−n ] = 1/2. Define
¨
σ(An : n ≥ 1)
for 0 ≤ t < 1
F t :=
σ(An , Zn : n ≥ 1) for t ≥ 1
P
completed with respect to P. Define Yn := 1≤k≤n Zk 1An and Tn := ∞1S1≤k≤n Ak .
Let X t be zero for t < 1 and Y∞ for t ≥ 1. Then (Tn )n≥1 is a localizing sequence
T
for X and X t n is zero for t < 1 and Yn for t ≥ 1. Since Yn is bounded for each n,
X Tn is a u.i. martingale, but X is not a martingale because X 1 = Y∞ ∈
/ L1.
1.7
Integration with respect to processes of finite variation
1.7.1 Definition. We say that a process A is increasing or of finite variation if it
has that property path-by-path for almost all ω.
1.7.2 Definition. Let A be of finite variation. The total variation process is
|A| t := sup
2n
X
n≥1 k=1
|A t(k+1)2−n − A t k2−n |.
Note that |A| t < ∞ a.s., |A| is increasing, and if A is adapted then so is |A|.
21
22
Preliminaries
1.7.3 Definition. Let A be of finite variation and F (ω, s) be (F ⊗ B)-measurable
and bounded.
Z
t
(F · A) t (ω) :=
F (s, ω)dAs (ω),
0
the path-by-path Lebesgue-Stieltjes integral (which is well-defined and finite for
almost all ω).
The integral process is also of finite variation and if A is (right) continuous
then the integral process is (right) continuous. If F is continuous path-by-path
then then integral may be taken to be the Riemann-Stieltjes integral.
1.7.4 Theorem. Let A and C be adapted, increasing processes such that C − A is
increasing. There exists H jointly measurable and adapted such that A = H · C.
PROOF (SKETCH): The correct process is
H(t, ω) :=
lim
r↑1,r∈Q+
A(ω, t) − A(ω, r t)
C(ω, t) − C(ω, r t)
.
ƒ
1.7.5 Corollary. If A is of finite variation then there is a jointly measurable process
H with −1 ≤ H ≤ 1 such that A = H · |A| and |A| = H · A.
PROOF (SKETCH): Let A+ := (|A| + A)/2 and A− := (|A| − A)/2. These are both
increasing processes, so by Theorem 1.7.4 there are H + and H − such that A+ =
H + · |A| and A− = H − · |A|. Take H := H + − H − .
ƒ
1.7.6 Theorem.
(i) Let A be of finite variation, H have continuous paths, and (Πn )n≥1 be a
sequence of partitions of [0, t] with mesh size converging to zero. For each
t i ∈ Πn let si denote any number in the interval [t i , t i+1 ]. Then
lim
n→∞
X
Πn
Hsi (A t i+1 − A t i ) =
Z
t
Hs dAs .
0
(ii) (Change of variable formula.) Let A be of finite variation with continuous
paths and f ∈ C 1 (R). Then ( f (A t )) t≥0 is of finite variation and
f (A t ) = f (A0 ) +
Z
t
f 0 (As )dAs .
0
(iii) Let A be of finite variation with continuous paths g be a continuous function.
Rt
R At
Then then 0 g(As )dAs = 0 g(s)ds.
1.8. Naïve stochastic integration is impossible
23
PROOF (OF (II)): Let (Πn )n≥1 be a sequence of partitions of [0, t] as in (i) and
consider
X
f (A t ) − f (A0 ) =
f (A t i+1 ) − f (A t i )
Πn
=
X
Πn
Z
→
f 0 (Asi )(A t i+1 − A t i )
for some si by the MVT
f 0 (As )dAs
by part (i)
t
ƒ
0
1.8
Naïve stochastic integration is impossible
1.8.1 Theorem (Banach-Steinhaus). Let X be a Banach space and let Y be a
normed linear space. Suppose (Tα )α∈I is a collection of continuous linear operators from X to Y such that supα∈I kTα xkY < ∞ for all x ∈ X . Then it is the case
that supα∈I kTα kL(X ,Y ) < ∞.
1.8.2 Theorem. Let x : [0, 1] → R be right continuous and let (Πn )n≥1 be a sequence of partitions of [0, 1] with mesh size converging to zero. If
X
h(t i )(x t i+1 − x t i )
Sn (h) :=
Πn
converges for all h ∈ C([0, 1]) then x is of finite variation.
PROOF: Note that Sn is a linear operator from C([0, 1]) to R, and
X
|x t i+1 − x t i |.
kSn k ≤
Πn
This upper bound is achieved for each n by the piecewise linear function hn with
the property that h(t i ) = sign(x t i+1 − x t i ). If Sn (h) exists for every h then the
Banach-Steinhaus theorem implies that supn≥1 kSn k < ∞, i.e. that the total variation of x over [0, 1] is finite.
ƒ
Remark. It should be possible to prove that if X is a right continuous process
and (ΠP
n )n≥1 is a sequence of partitions of [0, t] mesh size converging to zero
then if t i ,t i+1 ∈Πn H t i (X t i − X t i−1 ) converges in probability for all H ∈ F ⊗ B with
continuous paths then X is of finite variation a.s.
Chapter 2
Semimartingales and stochastic
integration
2.1
Introduction
2.1.1 Definition. A process H is a simple predictable process if it has the form
H t = H0 1{0} (t) +
n−1
X
H i 1(Ti ,Ti+1 ] (t)
i=1
where 0 = T0 ≤ T1 ≤ · · · ≤ Tn < ∞ are stopping times and H i ∈ L ∞ (F Ti ) for all
i = 0, . . . , n. The collection of all simple predictable processes will be denoted S. A
norm on S is kHku := sups≥0 kHs k∞ . When we endow S with the topology induced
by k · ku we write Su .
Remark. S ⊆ P .
2.1.2 Definition. L 0 := L 0 (Ω, F , P) := {X ∈ F : X is finite-valued a.s.} with the
topology induced by convergence in probability under P.
Remark. L 0 is not locally convex if P is non-atomic. The topological dual of L 0 in
this case is {0}.
2.1.3 Definition. Let H ∈ S have its canonical representation. For an arbitrary
stochastic process X , the stochastic integral of H with respect to X is
I X (H) := H0 X 0 +
n−1
X
H i (X Ti+1 − X Ti ).
i=1
A càdlàg adapted process X is called a total semimartingale if the map I X : Su → L 0
is continuous. X is a semimartingale if X t is a total semimartingale for all t ≥ 0.
24
2.2. Stability properties of semimartingales
2.2
25
Stability properties of semimartingales
2.2.1 Theorem.
(i) {semimartingales} and {total semimartingales} are vector spaces.
(ii) If Q P then any (total) semimartingale under P is also a (total) semimartingale under Q.
PROOF:
(i) Sums and scalar multiples of continuous operations are continuous.
(ii) Convergence in P-probability implies convergence in Q-probability.
ƒ
2.2.2 Theorem (Stricker). Let X be a semimartingale with respect to the filtration F and G is a sub-filtration of F (i.e. G t ⊆ F t for all t). If X is G adapted then
X is a semimartingale with respect to G.
PROOF: S(G) ⊆ S(F) and the restriction of a continuous operation is continuous.ƒ
2.2.3 Theorem. Let A = (Aα )α∈I be a collection of pairwise disjoint events from
F and let H t = σ(F t ∨ A). If X is a semimartingale with respect to F then X is a
semimartingale with respect to H.
Remark. This theorem is called Jacod’s countable expansion theorem in the textbook.
PROOF: Note first that H t = σ(F t ∨ (A \ {A ∈ A : P[A] = 0}) since F is complete,
and second that {A ∈ A : P[A] ≥ 1/n} is finite because the events are pairwise
disjoint. It follows that we may assume
S∞ without loss of generality that A is a
countable collection (An )n≥1 . If C := n=1 An then we may assume without loss of
generality that C C ∈ A, i.e. that A is a (countable) partition of Ω into events with
positive probability.
For each n define Qn := P[·|An ], so that Qn P. By Theorem 2.2.1 X is an
(F, Qn )-semimartingale. Let Jn be the filtration F completed with respect to Qn .
Claim. X is a (Jn , Qn )-semimartingale.
Indeed, if H ∈ S(Jn , Qn ) has its canonical representation then the Ti are Jn stopping times and the H i are in L ∞ (Ω, J Tni , Qn ).
Fact. Any Jn -stopping time is Qn -a.s. equal to a F-stopping time.
Denote the corresponding F-stopping times by Ti0 . Then any H i ∈ J Tni is Qn -a.s.
equal to an r.v. in F Ti0 (technical exercise). This shows that H is Qn -a.s. equal to a
process in S(F, Qn ), which proves the claim.
Note finally that Qn [Am ] ∈ {0, 1} for all Am ∈ A, so F ⊆ H ⊆ Jn for all n. By
Stricker’s theorem,
X is an (H, Qn )-semimartingale for all n. It can be shown that,
P∞
since dP = n=1 P[An ]dQn , X is an (H, P)-semimartingale.
ƒ
26
Semimartingales and stochastic integration
2.2.4 Theorem. Let X be a process and (Tn )n≥1 be a sequence of random times
such that X Tn is a semimartingale for all n and Tn ↑ ∞ a.s. Then X is a semimartingale.
PROOF: Fix t > 0. There is n0 such that P[Tn ≤ t] < " for all n ≥ n0 . Suppose that
H k ∈ S are such that kH k k∞ → 0. Let k0 be such that P[|I X Tn0 (H k )| > α] < " for
all k ≥ k0 . Then
P[|I X t (H k )| > α] = P[|I X t (H k )| > α; Tn0 ≤ t] + P[|I(X Tn0 )t (H k )| > α; Tn0 > t] < 2"
for all k ≥ k0 . Therefore I X t (H k ) → 0 in probability, so X is a semimartingale.
2.3
ƒ
Elementary examples of semimartingales
2.3.1 Theorem. Every adapted càdlàg process with paths of finite (total) variation is a (total) semimartingale.
PROOF: Suppose that the
R ∞total variation of X is finite. It is easy to see that, for all
H ∈ S, |I X (H)| ≤ kHk∞ 0 d|X |s .
ƒ
2.3.2 Theorem. Every càdlàg L 2 -martingale is a total semimartingale.
PROOF: Without loss of generality X 0 = 0. Let H ∈ S have the canonical representation.
2 X
n−1
2
E[(I X (H)) ] = E
H i (X Ti+1 − X Ti )
i=1
=E
X
n−1
H i2 (X Ti+1 − X Ti )2
X is a martingale
i=1
X
n−1
2
(X Ti+1 − X Ti )
≤
kHk2u E
=
i=1
2
kHku E[X T2n ]
≤
2
kHk2u E[X ∞
]
X is a martingale
by Jensen’s inequality
Therefore if H k → H in Su then I X (H k ) → I X (H) in L 2 , so also in probability.
ƒ
2.3.3 Corollary.
(i) A càdlàg locally square integrable martingale is a semimartingale.
(ii) A càdlàg local martingale with bounded jumps is a semimartingale.
(iii) A local martingale with continuous paths is a semimartingale.
(iv) Brownian motion is a semimartingale.
(v) If X = M + A is càdlàg, where M is a locally square integrable martingale
and A is locally of finite variation, then X is a semimartingale. Such an X is
called a decomposable process
(vi) Any Lévy process is a semimartingale by the Lévy-Itô decomposition.
2.4. The stochastic integral as a process
2.4
27
The stochastic integral as a process
2.4.1 Definition.
(i) D := {adapted, càdlàg processes}
(ii) L := {adapted, càglàd processes}
(iii) If C is a collection of processes then bC is the collection of bounded processes
from C.
2.4.2 Definition. For processes H n and H we say that H n → H uniformly on compacts in probability (or u.c.p.) if
h
i
lim P sup |Hsn − Hs | ≥ " = 0
n→∞
0≤s≤t
(p)
for all " > 0 and all t. Equivalently, if (H n − H)∗t −
→ 0 for all t.
Remark. If we define a metric
d(X , Y ) :=
∞
X
1
n=1
2n
E[(X − Y )∗n ∧ 1]
then u.c.p. convergence is compatible with this metric. We will denote by Ducp ,
Lucp , and Sucp the topological spaces D, L, and S endowed with the u.c.p. topology.
The u.c.p. topology is weaker than the uniform topology. It is important to note
that Ducp and Lucp are complete metric spaces.
2.4.3 Theorem. S is dense in Lucp .
PROOF: Let Y ∈ L and define R n := inf{t : |Yt | > n}. Then Y R n → Y u.c.p.
(exercise) and Y Rn is bounded by n because Y is left continuous. Thus bL is dense
in Lucp , so it suffices to show that S is dense in bLucp .
Assume that Y ∈ bL and define Z t := limu↓t Yn for all t ≥ 0, so that Z ∈ D. For
each " > 0 define a sequence of stopping times T0" := 0 and
"
Tn+1
:= inf{t > Tn" : |Z t − Z Tn" | > "}.
We have seen that the Tn" are stopping times because they are hitting times for the
càdlàg process Z. Also because Z ∈ D, Tn" ↑ ∞ a.s. (this would not necessarily
happen it T " were defined in the same way but with Y ). Let
X
"
Z " :=
Z Tn" 1[Tn" ,Tn+1
).
n≥0
Then Z " is bounded and Z " → Z uniformly on compacts as " → 0. Let
X
"
Z Tn" 1(Tn" ,Tn+1
U " := Y0 1{0} +
].
n≥0
28
Semimartingales and stochastic integration
Then U " → Y0 1{0} + Z− uniformly on compacts as " → 0. If we define
Y n," := Y0 1{0} +
n
X
"
Z Tk" 1(Tk" ,Tk+1
]
k≥1
then Y n," → Y u.c.p. as " → 0 and n → ∞.
ƒ
2.4.4 Definition. Let H ∈ S and X be a càdlàg process.
Pn The stochastic integral
process of H with respect to X is JX (H) := H0 X 0 + i=1 H i (X Ti+1 − X Ti ) when
Pn−1
H = H0 1{0} + i=1 H i 1(Ti ,Ti+1 ] .
R
Rt
Notation. We write Hs dX s := H ·X := JX (H) and 0 Hs dX s := I X t (H) = (JX (H)) t
R∞
and 0 Hs d X s := I X (H).
2.4.5 Theorem. If X is a semimartingale then JX : Sucp → Ducp is continuous.
PROOF: We begin by proving the weaker statement, that JX : Su → Ducp is continuous. Suppose that kH k k∞ → 0. Let δ > 0 and define a sequence of stopping times
T k := inf{t : |(H k · X ) t | ≥ δ}. Then H k 1[0,T k ] ∈ S and kH k 1[0,T k ] k∞ → 0 because
this already happens for (H k )k≥1 . Let t ≥ 0 be given.
P[(H k · X )∗t ≥ δ] ≤ P[(H k · X )∗t∧T k ≥ δ] = P[I X t (H k 1[0,T k ] ) ≥ δ] → 0
since X is a semimartingale. Therefore JX (H k ) → 0 u.c.p.
Assume H k → 0 u.c.p. Let " > 0, t > 0, and δ > 0 be given. There is η such
that kHk∞ < η implies P[JX (H)∗t ≥ δ] < ". Let Rk := inf{s : |Hsk | > η}. Then
Rk → ∞ a.s., H̃ k := H k 1[0,Rk ] → 0 u.c.p., and kH̃ k k ≤ η.
P[(H k · X )∗t ≥ δ] = P[(H k · X )∗t ≥ δ; Rk < t] + P[(H k · X )∗t ≥ δ; Rk ≥ t]
≤ P[Rk < t] + P[(H̃ k · X )∗t ≥ δ] < 2"
for k large enough.
ƒ
If H ∈ L and (H n )n≥1 ⊆ S are such that H n → H u.c.p. then (H n )n≥1 has the
Cauchy property for the u.c.p. metric. Since JX is continuous, (JX (H n ))n≥1 also
has the Cauchy property. Since Ducp is complete there is JX (H) ∈ D such that
JX (H n ) → JX (H) u.c.p. We say that JX (H) is the stochastic integral of H with
respect to X .
2.4.6 Example. Let B be standard Brownian motion and let (Πn )n≥1 be a sequence of partitions of [0, t] with mesh size converging to zero. Define
X
B n :=
B t i 1(t i ,t i+1 ] ,
Πn
2.5. Properties of the stochastic integral
29
so that B n → B u.c.p. (check this as an exercise).
X
(JB (B n )) t =
B t i (B t i+1 − B t i )
Πn
1X 2
1X
=
(B t i+1 − B 2t i ) −
(B t i+1 − B t i )2
2 Π
2 Π
n
n
1 2 1X
(B t i+1 − B t i )2
= Bt −
2
2 Π
n
(p)
−
→
1
2
B 2t −
1
2
t
Hence
Rt
2.5
Properties of the stochastic integral
0
Bs d Bs = 12 Bs − 21 t.
2.5.1 Theorem. Let H ∈ L and X be a semimartingale.
(i) If T is a stopping time then (H · X ) T = (H1[0,T ] · X ) = H · (X T ).
(ii) ∆(H · X ) is indistinguishable from H · (∆X ).
(iii) If Q P
P then H ·Q X is indistinguishable
from H ·P X .
P
(iv) If Q = k≥1 λk Pk with λk ≥ 0 and k≥1 λk = 1 then H ·Q X is indistinguishable from H ·Pk X for any k for which λk > 0.
(v) If X is both a P and Q semimartingale then there is H · X that is a version of
both H ·P X and H ·Q X .
(vi) If G is another filtration and H ∈ L(G) ∩ L(F) then H ·G X = H ·F X .
(vii) If X has paths of finite variation on compacts then H · X is indistinguishable
from the path-by-path Lebesgue-Stieltjes integral.
(viii) Y := H · X is a semimartingale and K · Y = (K H) · X for all K ∈ L.
PROOF (OF (VIII)): It can be shown using limiting arguments that K · (H · X ) =
(K H) · X when H, K ∈ S. To prove that Y = H · X is a semimartingale when H ∈ L
there are two steps. First we show that if K ∈ S then K · Y = (K H) · X (and this
makes sense even without knowing that Y is a semimartingale). Fix t > 0 and
choose H n ∈ S with H n → H u.c.p. We know H n · X → H · X u.c.p. because X is a
semimartingale. Therefore there is a subsequence such that (Y nk − Y )∗t → 0 a.s.,
where Y nk := H nk · X . Because of the a.s. convergence, (K · Y ) t = limk→∞ (K · Y nk ) t
a.s. Finally, K · Y nk = (K H nk ) · X since H n , K ∈ S, so
K · Y = u.c.p.-lim (K H nk ) · X = (K H) · X
k→∞
since K H nk → K H u.c.p. and X is a semimartingale.
For the second step, proving that Y is a semimartingale, assume that Gn → G
in Su . We must prove that (G n · Y ) t → (G · Y ) t in probability for all t. We have
G n H → GH in Lucp , so
lim G n · Y = lim (G n H) · X = (GH) · X = G · Y.
n→∞
n→∞
Since this convergence is u.c.p. this proves the result.
ƒ
30
Semimartingales and stochastic integration
2.5.2 Theorem. If X is a locally square integrable martingale and H ∈ L then
H · X is a locally square integrable martingale.
PROOF: It suffices to prove the result assuming X is a square integrable martingale,
since if (Tn )n≥1 is a localizing sequence for X then (H · X ) Tn = H · X Tn , so we would
have that H · X is a local locally square integrable martingale. Conclude with
Theorem 1.6.2, since the collection of locally square integrable martingale is stable
under stopping. Furthermore, we may assume that H is bounded because left
continuous processes are locally bounded (a localizing sequence is R n := inf{t :
|H t | > n}) and that X 0 = 0.
Construct, as in Theorem 2.4.3, H n ∈ S such that H n → H u.c.p. By construction the H n are bounded by kHk∞ . Let t > 0 be given. It is easy to see that H n · X
is a martingale and
2
E[(H · X ) ] = E
n
X
n
H in
T
X t i+1
2 −
T
Xt i
i=1
≤ kHk2∞ E[X t2 ] < ∞.
(The detail are as in the proof of Theorem 2.3.2.) It follows that ((H n · X ) t )n≥1 is
u.i. Since we already know (H n · X ) t → (H · X ) t in probability, the convergence is
actually in L 1 . This allows us to prove that H · X is a martingale, and by Fatou’s
lemma
E[(H · X )2t ] ≤ lim inf E[(H n · X )2t ] ≤ kHk2∞ E[X t2 ] < ∞
n
so H · X is a square integrable martingale.
ƒ
2.5.3 Theorem. Let X be a semimartingale. Suppose that (Σn )n≥1 is a sequence
of random partitions, Σn : 0 = T0n ≤ T1n ≤ · · · ≤ Tkn where the Tkn are stopping
n
times such that
(i) limn→∞ Tkn = ∞ a.s. and
n
n
(ii) supk |Tk+1
− Tkn | → 0 a.s.
R
P
n
n
Then if Y ∈ L (or D) then k YTkn (X Tk+1 − X Tk ) → Y− dX .
Remark. The hard part of the proof of this theorem is that the approximations to
Y do not necessarily converge u.c.p. to Y (if they did then the theorem would be
a trivial consequence of the definition of semimartingale).
2.5.4 Example. Let M t = Nt − λt be a compensated Poisson process (a martingale) and let H = 1[0,T1 ) (a bounded process in D), where T1 is the first jump time
Rt
of the Poisson process. Then 0 Hs d Ms = −λ(t ∧ T1 ), which is not a local martingale. Examples of this kind are part of the reason that we want our integrands
from L.
2.6. The quadratic variation of a semimartingale
2.6
The quadratic variation of a semimartingale
2.6.1 Definition. Let X and Y be semimartingales. The quadratic variation of X
and Y is [X , Y ] = ([X , Y ] t ) t≥0 , defined by
Z
Z
[X , Y ] := X Y −
X − dY −
Y− dX .
Write [X ] := [X , X ]. The polarization identity is sometimes useful.
[X , Y ] =
1
2
([X + Y ] − [X ] − [Y ])
2.6.2 Theorem.
(i) [X ]0 = X 02 and ∆[X ] = (∆X )2 .
(ii) If (Σn )n≥1 are as in Theorem 2.5.3 then
X
n
n
u.c.p.
(X Tk+1 − X Tk )2 −→ [X ].
X 02 +
k
(iii) [X ] is càdlàg, adapted, and increasing.
(iv) [X , Y ] is of finite variation, [X , Y ]0 = X 0 Y0 , ∆[X , Y ] = ∆X ∆Y , and
X
n
n
n
n
u.c.p.
X 0 Y0 +
(X Tk+1 − X Tk )(Y Tk+1 − Y Tk ) −→ [X , Y ].
k
(v) If T is any stopping time then
[X T , Y ] = [X , Y T ] = [X T , Y T ] = [X , Y ] T .
PROOF:
R0
(i) Recall that X 0− := 0, so 0 X s− dX s = 0 and [X ]0 = (X 0 )2 . For any t > 0,
2
− 2X t X t−
(∆X t )2 = (X t − X t− )2 = X t2 + X t−
2
= X t2 − X t−
− 2X t− ∆X t
= (∆X 2 ) t − 2∆(X − · X ) t
= ∆(X 2 − 2(X − · X )) t = ∆[X ] t
(ii) Fix n and let R n := supk Tkn . Then (X 2 )Rn → X 2 u.c.p., so apply Theorem 2.5.3 and the summation-by-parts trick.
(iii) Let’s see why [X ] is increasing. Fix s < t rational. If we can prove that
[X ]s ≤ [X ] t a.s. then we are done since [X ] is càdlàg. Use partitions (Σn )n≥1
in Theorem 2.5.3 that include s and t. Excluding terms from the sum makes
it smaller since all of the summands are squares (hence non-negative), so
[X ]s ≤ [X ] t a.s.
ƒ
31
32
Semimartingales and stochastic integration
2.6.3 Corollary.
(i) If X is a continuous semimartingale of finite variation then [X ] is constant,
i.e. [X ] t = X 02 for all t.
(ii) [X , Y ] and X Y are semimartingales, so {semimartingales} form an algebra.
2.6.4 Theorem (Kunita-Watanabe identity). Let X and Y be semimartingales and
H, K : Ω × [0, ∞) → R be F ⊗ B-measurable.
Z∞
Z ∞
2 Z ∞
2
|Hs ||Ks |d|[X , Y ]|s
≤
Hs d[X ]s
Ks2 d[Y ]s .
0
0
0
PROOF: Use the following lemma.
2.6.5 Lemma. Let α, β, γ : [0, ∞) → R be such that α(0) = β(0) = γ(0) = 0, α is
of finite variation, β and γ are increasing, and for all s ≤ t,
(α(t) − α(s))2 ≤ (β(t) − β(s))(γ(t) − γ(s)).
Then for all measurable functions f and g and all s ≤ t,
Z t
Z t
2 Z t
2
| fu ||gu |d|α|u
≤
fu dβu
gu2 dγu .
s
s
s
The lemma can be proved with the following version of the monotone class theorem.
2.6.6 Theorem (Monotone class theorem II). Let C be an algebra of bounded
real valued functions. Let H be a collection of bounded real valued functions
closed under monotone and uniform convergence. If C ⊆ H then bσ(C) ⊆ H.
Both are left as exercises.
ƒ
2.6.7 Definition. Let X be a semimartingale. The continuous part of quadratic
variation, [X ]c , is defined by
X
[X ] t = [X ]ct + X 02 +
(∆X s )2 .
0<s≤t
X is said to be a quadratic pure jump semimartingale if [X ]c ≡ 0.
2.6.8 Examples.
(i) If N is a Poisson process then [N ] = N , so it is a quadratic pure jump
semimartingale.
(ii) Any Lévy process with no Gaussian part is a quadratic pure jump semimartingale. Note that [t] ≡ 0.
2.6.9 Theorem. If X is an adapted, càdlàg process of finite variation then X is a
quadratic pure jump semimartingale.
2.6. The quadratic variation of a semimartingale
33
PROOF: The Lebesgue-Stieltjes integration-by-parts formula gives us
Z
Z
X2 =
X − dX +
X dX ,
and by definition of [X ] we have
2
X =2
Z
X − dX + [X ].
Noting that
Z
we obtain [X ] =
X dX =
Z
(X − + ∆X )dX =
Z
X − dX +
X
(∆X )2
P
(∆X )2 , so X is a quadratic pure jump semimartingale.
ƒ
2.6.10 Theorem. If X is a locally square integrable martingale that is not constant
everywhere then [X ] is not constant everywhere. Moreover, X 2 − [X ] is a local
martingale and if [X ] ≡ 0 then X ≡ 0.
Remark. This theorem holds when X is a local martingale, but it is harder to
prove.
R
PROOF: We have that X 2 − [X ] = 2 X − dX is a locally square integrable martingale by Theorem 2.5.2. Assume that [X ] t = 0 for all t ≥ 0. Then X 2 is a locally
square integrable martingale, so suppose that (Tn )n≥1 is a localizing sequence.
2
2
Then E[X t∧T
] = X 02 = 0 for all n and all t. Thus X t∧T
= 0 a.s. for all n and all t,
n
n
so X ≡ 0. If X 0 is a constant other than zero then the proof applies to the locally
square integrable martingale X − X 0 .
ƒ
2.6.11 Corollary. Let X be a locally square integrable martingale and S ≤ T be
stopping times.
(i) If [X ] is constant on [S, T ] ∩ [0, ∞) then X is constant there too.
(ii) If X has continuous paths of finite variation on (S, T ) then X is constant on
[S, T ] ∩ [0, ∞).
PROOF: Consider M := X T − X S and apply Theorems 2.6.2 and 2.6.10.
ƒ
2.6.12 Corollary.
(i) If X and Y are locally square integrable martingales then [X , Y ] is the
unique adapted, càdlàg process of finite variation such that
a) X Y − [X , Y ] is a local martingale, and
b) ∆[X , Y ] = ∆X ∆Y and [X , Y ]0 = X 0 Y0 .
(ii) If X and Y are locally square integrable martingales with no common jumps
and such that X Y is a local martingale then [X , Y ] ≡ X 0 Y0 .
34
Semimartingales and stochastic integration
(iii) If X is a continuous square integrable martingale and Y is a square integrable martingale of finite variation then [X , Y ] ≡ X 0 Y0 and X Y is martingale.
Fill in more of this proof.
PROOF:
R
R
(i) By definition, X Y − [X , Y ] = X − dY + Y− dX , which is a local martingale.
The second condition is Theorem 2.6.2(vi). Conversely, if A satisfied the
two conditions then, by Theorem 2.6.2(vi), A − [X , Y ] is a continuous semimartingale of finite variation null at zero. By Corollary 2.2.3, A = [X , Y ]. ƒ
2.6.13 Assumption. [M ] always exists for a local martingale M .
2.6.14 Corollary. Let M be a local martingale. Then M is a square integrable
martingale if and only if E[[M ] t ] < ∞ for all t ≥ 0. We have E[M t2 ] = E[[M ] t ]
when E[[M ] t ] < ∞.
PROOF: Suppose M is a square integrable martingale. M 2 − [M ] is a local martingale null at zero by Corollary 2.6.12. Let (Tn )n≥1 be a localizing sequence. For all
t ≥ 0,
2
] ≤ E[M t2 ] < ∞
E[[M ] t ] = lim E[[M ] t∧Tn ] = lim E[M t∧T
n
n→∞
n→∞
The first equality is the monotone convergence theorem, the second is because
(M 2 −[M ]) Tn ∧t is a martingale, and the inequality is the “limsup” version of Fatou’s
lemma. For the converse, define stopping times Tn := inf{t : |M t | > n} ∧ n.
p
sup |MsTn | ≤ n + |∆M Tn | ≤ n + [M ]n
ƒ
0≤s≤t
Hence sups≤t |MsTn | ∈ L 2 ⊆ L 1 for all n and all t. By Doob’s maximal inequality. . .
2.6.15 Example (Inverse Bessel process). Let B be a three dimensional Brownian motion that starts at x 6= 0. It can be shown that M t := 1/kB t k is a local
martingale. Furthermore, E[M t2 ] < ∞ for all t and lim t→∞ E[M t2 ] = 0. This
implies that M is not a martingale (because if it were then M 2 would be a submartingale, which contradicts that lim t→∞ E[M t2 ] = 0). Moreover, E[[M ] t ] = ∞
for all t > 0. It can be shown that M satisfies the SDE d M t = −M t2 dB t .
2.6.16 Corollary. Let X be a continuous local martingale. Then X and [X ] have
the same intervals of constancy a.s. (i.e. pathwise, cf. Corollary 2.6.11).
2.6.17 Theorem. Let X be a quadratic pure jump semimartingale. For any semimartingale Y ,
X
[X , Y ] = X 0 Y0 +
∆X s ∆Ys .
0<s≤t
2.7. Itô’s formula
35
PROOF (SKETCH): We know by the Kunita-Watanabe inequality that d[X , Y ]s d[X , X ]s a.s. Since [X , X ]c ≡ 0, this implies that [X , Y ]c ≡ 0, so [X , Y ] is equal to
the sum of its jumps.
ƒ
2.6.18 Theorem. Let H, K ∈ L and X and Y be semimartingales.
(i) [H · X , K · Y ] = (H K) · [X , Y ].
(ii) If H is càdlàg and (σn )n≥1 is a sequence of random partitions tending to the
identity then
Z
X
n
n
Tin u.c.p.
Ti+1
Tin
Ti+1
− Y ) −→ Hs− d[X , Y ]s .
H Tin (X
− X )(Y
Remark. Part (i) is important for computations, but part (ii) is important theoretically. We will prove Itô’s formula using part (ii).
PROOF (OF (I)): It is enough to prove that [H · X , Y ] = H · [X , Y ] since we already
have associativity. Suppose that H n ∈ S are such that H n → H u.c.p. Define
Z n := H n · X and Z := H · X , so that Z n → Z u.c.p. Then
Z
Z
[Z n , Y ] = Y Z n −
=YZ −
Y− d Z n −
Z
n
Z
→YZ −
Z−n dY
(Y H )− dX −
n
(Y H)− dX −
Z
Z−n dY
Z
Z− dY
= [Z, Y ]
For all n, since H n is simple predictable,
[Z n , Y ] = [H n · X , Y ] = H n [X , Y ]
(check this). Finally, [X , Y ] is a semimartingale so H n [X , Y ] → H · [X , Y ].
2.7
Itô’s formula
We have seen that if A is continuous and of finite variation and if f ∈ C 1 then
f (A t ) = f (A0 ) +
Z
t
f 0 (As )dAs .
0
We also saw that if B is a standard Brownian motion then
Z t
B 2t = 2
Bs dBs + t.
0
ƒ
36
Semimartingales and stochastic integration
Take f (x) = x 2 so that we can write
Z t
f (B t ) − f (B0 ) =
0
f (Bs )dBs +
0
1
2
t
Z
f 00 (Bs )d[B]s .
0
This expression does not coincide with the one for continuous processes of finite
variation.
Notation. We write
Z
Z
t
t
Z
Hs d X s :=
Hs dX s − H0 X 0 =
Hs dX s :=
(0,t]
0+
Z
0
t
Hs 1(0,∞) (s)dX s .
0
2.7.1 Theorem (Itô’s formula). If X is a semimartingale and f ∈ C 2 (R) then
( f (X t )) t≥0 is a semimartingale and
f (X t ) − f (X 0 ) =
Z
t
0
f (X s− )dX s +
0+
1
2
Z
t
f 00 (X s− )d[X ]sc
0+
X
+
( f (X s ) − f (X s− ) − f 0 (X s− )∆X s ).
0<s≤t
In particular, if X has continuous paths then
Z t
f 0 (X s )dX s +
f (X t ) − f (X 0 ) =
0+
1
2
Z
t
f 00 (X s )d[X ]s .
0
Additionally, if X is of finite variation then
Z t
X
f (X t ) − f (X 0 ) =
f 0 (X s− )dX s +
( f (X s ) − f (X s− ) − f 0 (X s− )∆X s ),
0+
0<s≤t
and this last formula holds even if f is only C 1 .
PROOF: First assume that X is bounded by a constant
P K. By definition the quadratic
variation of a semimartingale is finite valued, so 0<s≤t (∆X s )2 ≤ [X ] t < ∞ a.s.
Since
f 00 is continuous on the interval [−K, K] it is bounded on that interval, so
P
00
2
0<s≤t f (X s− )(∆X s ) < ∞ a.s. and we may write Itô’s formula in the following
equivalent form.
f (X t ) − f (X 0 ) =
Z
t
0
f (X s− )dX s +
0+
+
X
1
2
Z
t
f 00 (X s− )d[X ]s
0+
( f (X s ) − f (X s− ) − f 0 (X s− )∆X s − f 00 (X s− )(∆X s )2 )
0<s≤t
Fix t > 0. Taylor’s theorem states that, for x, y ∈ [−K, K],
f ( y) − f (x) = f 0 (x)( y − x) +
1
2
f 00 (x)( y − x)2 + R(x, y),
2.7. Itô’s formula
37
where the remainder satisfies |R(x, y)| ≤ r(|x − y|)(x − y)2 , where r is an increasing function
P such that lim"↓0 r(") = 0.
Since 0<s≤t (∆X s )2 is a.s. a convergent series, we can partition the jumps of
X in [0, t] intoPa finite set A := A(", t) and the remainder B := B(", t), with the
property that s∈B (∆X s )2 ≤ " 2 . A are the times of “large” jumps and B are the
times of “small” jumps.
Suppose that (Σn )n≥1 is a sequence of random dyadic partitions of [0, t] with
mesh size converging to zero, say Σn : (0 = T0n ≤ T1n ≤ · · · ≤ Tkn = t). For each n,
n
partition the indices {0, 1, . . . , kn − 1} into the follow sets
n
Ln := {i : (Tin , Ti+1
] ∩ A 6= ∅}
n
Sn := {i ∈
/ Ln : (Tin , Ti+1
] ∩ B 6= ∅}
Rn := {0, 1, . . . , kn − 1} \ (Ln ∪ Sn )
Then Ln is the collection of indices i for which the i th interval in the partition
contains a large jump of X , Sn is the collection of indices i for which the i th interval
contains a small jump but no large jump, and Sn is the collection of indices i for
which X is continuous on the i th interval. Write
X
n ) − f (X n ))
f (X t ) − f (X 0 ) =
( f (X Ti+1
Ti
Σn
=
X
Ln
=: S Ln
+
X
Sn
+ SSn
+
X
n ) − f (X n ))
( f (X Ti+1
Ti
Rn
+ SRn
By Taylor’s theorem we can write
SSn + SRn =
X
Σn
n
f 0 (X Tin )(X Ti+1
− X Tin ) +
−
X
Ln
1
2
n
− X Tin )2
f 00 (X Tin )(X Ti+1
n
f 0 (X Tin )(X Ti+1
− X Tin ) +
+
X
S n ∪R n
1
2
n
f 00 (X Tin )(X Ti+1
− X Tin )2
n ).
R(X Tin , X Ti+1
P
P
n
As n → ∞, Sn (X Ti+1
− X Tin )2 → s∈B (∆X s )2 . The remainder term in the above
expression, call it Rn , satisfies
X
n
n )
|R | = R(X Tin , X Ti+1
S n ∪R n
X
2
n |)(X n
r(|X Tin − X Ti+1
≤
Ti+1 − X Tin )
S n ∪R n
=
X
+
X
Sn
≤ r(2K)
Rn
2
n |)(X n
r(|X Tin − X Ti+1
Ti+1 − X Tin )
X
X
n
n
n
(X Ti+1
− X Tin )2 + sup r(|X Ti+1
− X Tin |)
(X Ti+1
− X Tin )2
Sn
Rn
Rn
38
Semimartingales and stochastic integration
By the properties of r we can pick a subsequence (Σnm )m≥1 such that
For each m ≥ 1 there is nm such that
X
X
1
n 2
2
n
(X Ti+1
− XTn) −
(∆X s ) ≤ 1 +
"2 .
i
m
x∈B
σ nm ∩B6=0
You can find a sequence of partitions σ̃ m , constructed by refining σ nm outside the
subintervals that don’t contain jumps, such that
X
m ) → 0
R(X T̃im , X T̃i+1
σ̃ m ∩A=∅
as m → ∞ and " → 0. Work from then on with the refinements σ̃ m .
lim sup |Rnm | ≤ (1 + r(2K))" 2 .
m→∞
Hence
n
n
n
f (X t ) − f (X 0 ) = S L m + SS m + SR m
X
1
n
n
n
− X Tin )2
− X Tin ) + f 00 (X Tin )(X Ti+1
= SLm +
f 0 (X Tin )(X Ti+1
2
Σnm
X
1
n
n
−
f 0 (X Tin )(X Ti+1
− X Tin ) + f 00 (X Tin )(X Ti+1
− X Tin )2 + Rnm
2
n
L
As m → ∞,
X
0
f (X )(X
Σnm
X1
Σnm
2
Tin
n
Ti+1
−X )→
Z
t
f 0 (X s− )dX s
Tin
by Theorem 2.5.3
0+
00
2
n
− X Tin ) →
f (X Tin )(X Ti+1
1
2
Z
t
f 00 (X s− )d[X ]s
by Theorem 2.6.18
0+
and similarly,
X
S Ln =
Ln
X
Ln
X1
Ln
2
n ) − f (X n ) →
f (X Ti+1
Ti
n
f 0 (X Tin )(X Ti+1
− X Tin ) →
n
f 00 (X Tin )(X Ti+1
− X Tin )2 →
X
f (X s ) − f (X s− )
s∈A
X
f 0 (X s− )∆X s
s∈A
X1
2
s∈A
f 00 (X s− )(∆X s )2
Finally, as " → 0, A(", t) exhausts the jumps of X . Since
f (X s ) − f (X s− ) − f 0 (X s− )∆X s +
1
2
f 00 (X s− )(∆X s )2
is dominated by a constant times the sum of the jumps squared. . .
2.7. Itô’s formula
39
If X is not necessarily bounded then let Vk := inf{t : |X t | > k} and Y k :=
X 1[0,Vk ) . Then Y k is a semimartingale, as it is a product of two semimartingales,
bounded by k. (Note that Y k is not the same as X Vk − , as the former jumps to zero
at time Vk .) By the previous part,
f
(Ytk ) −
f
(Y0k )
=
Z
t
f
0
k
(Ys−
)d Ysk
+
0+
t
Z
1
2
k
f 00 (Ys−
)d[Y k ]sc
0+
X
+
k
k
( f (Ysk ) − f (Ys−
) − f 0 (Ys−
)∆Ysk ).
0<s≤t
Fix ω and t and k such that Vk (ω) > t. One can rewrite the above in terms of
Itô’s formula for X . (More perspicaciously, for fixed t, for each ω there is k large
enough so that Y k (ω, t) = X (ω, t) and the differentials are equal too.)
ƒ
2.7.2 Theorem.
(i) If X = (X 1 , . . . , X n ) is an n-dimensional semimartingale and f : Rn → R has
continuous second order partial derivatives then f (X ) = ( f (X t1 , . . . , X tn )) t≥0
is a semimartingale and
f (X t )− f (X 0 ) =
n
X
Z
i=1
t
0+
∂f
∂ xi
+
(X s− )dX si +
X n
1X
2
i=1
Z
t
0+
∂2f
∂ xi∂ x j
f (X s ) − f (X s− ) −
0<s≤t
(X s− )d[X i , X j ]sc
n
X
∂f
i=1
∂ xi
(X s− )∆X si
(ii) If X and Y are semimartingales, Z := X + iY , and f is analytic then
f (Z t ) − f (Z0 ) =
Z
t
0+
0
1
Z
t
f 00 (Zs− )d[Z]sc
2 0+
X
( f (Zs ) − f (Zs− ) − f 0 (Zs− )∆Zs ).
+
f (Zs− )d Zs +
0<s≤t
where “d Z = d X + id Y ” and [Z] = [X ] − [Y ] + 2i[X , Y ].
2.7.3 Definition. The Fisk-Stratonovich integral of is defined for semimartingales
Rt
Rt
X and Y by (Y− ◦ X ) t := 0 Ys− ◦ d X s := 0 Ys− dX s + 21 [X , Y ]ct .
2.7.4 Theorem.
(i) If X is a semimartingale and f ∈ C 3 (R)
Z t
X
f (X t ) − f (X 0 ) =
f 0 (X s− ) ◦ dX s +
( f (X s ) − f (X s− ) − f 0 (X s− )∆X s ).
0+
0<s≤t
(ii) If X and Y are semimartingales and at least one is continuous then
X Y − X 0 Y0 = X − ◦ Y + Y− ◦ X .
40
Semimartingales and stochastic integration
2.8
Applications of Itô’s formula
2.8.1 Theorem (Stochastic exponential).
Let X be a semimartingale with X 0 = 0. Then there exists a unique semimartingale
Rt
Z such that Z t = 1 + 0 Zs− dX s . Moreover,
Z t = E (X ) t := exp(X t − 12 [X ]ct )
Y
(1 + ∆X s ) exp(−∆X s ).
s≤t
PROOF: Let’s see first that E (X ) is a semimartingale. We know that X − 12 [X ]c is
a semimartingale, so exp(X − 21 [X ]c ) is a semimartingale by Itô’s theorem. We
Q
need to show that s≤t (1 + ∆X s )e−∆X s is well-defined and is a semimartingale.
Towards this, let U t = ∆X s 1|∆X s |≤ 1 . The set {s : |∆X s | > 12 } is finite a.s., so it is
2
Q
enough to prove that s≤t (1 + Us )e−Us is a semimartingale. For |x| ≤ 12 , it can be
shown that | log(1 + x) − x| ≤ x 2 , whence
X
Y
−Us log
| log(1 + Us ) − Us |
(1
+
U
)e
s
≤
s≤t
s≤t
X
X
≤
Us2 ≤
(∆X s )2 ≤ [X ] t < ∞ a.s.
s≤t
s≤t
Q
−∆X s
Hence the infinite product converges, and moreover,
is of
s≤t (1 + ∆X s )e
finite variation because the series of its log-jumps is absolutely summable. In
particular it is a quadratic pure jump semimartingale.
Now we prove that Z Q
satisfied the integral equation. Let F (x, y) := y e x ,
1
c
K t := X − 2 [X ] , and S t := s≤t (1 + ∆X s )e−∆X s . Then F is C 2 and Z t = F (K t , S t ).
By Itô’s formula,
Z t
Z
Z t
1
Zt − 1 =
Zs− d[K]sc
Zs− dKs +
e Ks− dSs +
2
0+
0+
X 0+
+
(Zs − Zs− − Zs− (∆Ks ) − e Ks− ∆Ss )
0<s≤t
=
Z
t
X
Zs− dKs +
0
e
Ks−
(∆Ss ) +
0<s≤t
+
X
1
Z
2
Zs− d[X ]sc
0
(Zs − Zs− − Zs− (∆X s ) − e Ks− ∆Ss )
0<s≤t
=
Z
0
t
Zs− dKs +
1
2
Z
Zs− d[X ]sc
0
=
Z
t
Zs− dX s
0
The only tricky point is to note that Zs = Zs− (1 + ∆X s ).
Remark.
(i) If X is of finite variation then so is E (X ).
ƒ
2.8. Applications of Itô’s formula
41
(ii) If X is a local martingale then so is E (X ), but at this point we can only prove
it if X is a locally square integrable martingale.
(iii) Let T = inf{t : ∆X t = −1}.
a) E (X ) 6= 0 on [0, T )
b) E (X )− 6= 0 on [0, T ]
c) E (X ) = 0 on [T, ∞)
(iv) If X is continuous and X 0 = 0 then E (X ) = exp(X − 12 [X ]).
(v) If λ ∈ R and B is standard Brownian motion then E (λB) = exp(λB t − 12 λ2 t)
is a continuous martingale.
2.8.2 Theorem (Yor’s formula). If X and Y are semimartingales with X 0 = Y0 =
0 then E (X ) E (Y ) = E (X + Y + [X , Y ]).
PROOF: Integration-by-parts.
ƒ
Remark. If X is continuous then (E (X ))−1 = E (−X + [X ]).
2.8.3 Theorem. Let X = (X 1 , . . . , X n ) be an n-dimensional local martingale with
values in D, an open subset of Rn . Assume [X i , X j ] = Aδi, j , where A is some
increasing process. If u : D → R is harmonic (resp. sub-harmonic) then u(X ) is a
local martingale (resp. sub-martingale).
2.8.4 Theorem (Lévy’s characterization of BM).
A stochastic process X is a Brownian motion if and only if X is a continuous local
martingale with [X ] t = t a.s. for all t.
PROOF: Only one direction requires proof, so assume X is a continuous local mar1 2
tingale and [X ] t = t for all t. It suffices to show that E[e iu(X t −X s ) | Fs ] = e− 2 u (t−s)
for all 0 ≤ s < t < ∞ (exercise). Fix u ∈ R, let F (x, t) := exp(iux + 12 u2 t), and let
Z t := F (X t , t). By Itô’s formula,
Z t − Z0 = iu
Z
t
Zs d X s +
0
Z t = 1 + iu
Z
u2
2
Z
t
Zs ds −
0
u2
Z
2
t
Zs d[X ]s
0
t
Zs dX s
0
Therefore Z is a local martingale by Theorem 2.5.2. But
1
2
sup |Z t | = sup | exp(iuX t + 12 u2 t)| = e 2 u t < ∞.
s≤t
s≤t
so Z is a martingale, and this implies what we wanted to prove.
ƒ
42
Semimartingales and stochastic integration
2.8.5 Theorem (Multidimensional Lévy’s characterization).
If X = (X 1 , . . . , X n ) is a vector of n continuous local martingales and, for all i, j,
[X i , X j ] t = tδi, j for all t then X is an n-dimensional Brownian motion.
2.8.6 Theorem. Let M be a continuous local martingale with M0 = 0 and such
that lim t→∞ [M ] t = ∞. Let
(i) Ts = inf{t : [M ] t > s},
(ii) Gs := F Ts and G = (Gs ) g≥0 , and
(iii) Bs := M Ts .
Then B is a G-Brownian motion, [M ] t is G-stopping time for all t, and M t = B[M ] t
a.s. for all t.
PROOF: Since [M ]∞ = ∞, Ts < ∞ for all s and B is well-defined. Also, since
Ts ≤ Tu for s ≤ u, F Ts = Gs ⊆ F Tu = Gu , so G really is a filtration. It is right
continuous because T is right continuous. For all s and t,
{[M ] t ≤ s} = {Ts ≥ t} ∈ F Ts = Gs ,
Ts
so the [M ] t are G-stopping times. E[[M ]∞
] = E[[M ] Ts ] = s for all s since [M ] is
a continuous process. By Corollary 1.4.8, M Ts is a u.i. F-martingale with E[M T2s ] =
E[[M ] Ts ] = s. By the optional sampling theorem
E[Bs |Gu ] = E[M Ts |F Tu ] = M Tu = Bu
for all u ≤ s, so B is a G-martingale. One can prove that (M Ts )2 − [M ] Ts is a u.i.
martingale for each s (exercise). Hence
E[Bs2 − Bu2 |Gu ] = E[M T2s − M T2u |G Tu ] = E[[M ] Ts − [M ] Tu |G Tu ] = s − u.
This implies that (Bu2 − u)u≥0 is a G-martingale. By Corollary 2.6.16 M and [M ]
have the same intervals of constancy, so B = M T is a continuous process. Corollary 2.6.12 implies that [B]u = u, so B is a Brownian motion by Lévy’s characterization. (See the exercises around p.99 in the text regarding time changes.)
Finally, note that T[M ] t ≥ t because T is the right continuous inverse of [M ], and
T[M ] t > t if and only if [M ] is constant on the interval (t, T[M ] t ), which happens if
and only if M is constant on that interval. Hence B[M ] t = M t for all t.
ƒ
2.8.7 Exercise.
then B is also a G̃-Brownian motion,
T If B is a G-Brownian motion
T
where G˜s = u>s Gu . Note that E[Z| u>s Gu ] = limu↓s E[Z|Gu ] (this equality is
known as Lévy’s theorem).
2.8.8 Theorem (Lévy’s stochastic area formula).
Let B
Rt
R t= (X , Y ) be a standard
2-dimensional Brownian motion and let A t := 0 X s dYs − 0 Ys dX s . Then E[e iuAt ] =
1/ cosh(ut) for all u and 0 ≤ t < ∞.
Chapter 3
The Bichteler-Dellacherie Theorem
and its connexions to arbitrage
3.1
Introduction
Classical semimartingales
3.1.1 Definition. An adapted càdlàg process S is a classical semimartingale if it
can be decomposed as S = S0 + M +A, where M0 = A0 = 0, M is a local martingale,
and A is of finite variation.
Remark. Being a classical semimartingale is a local property, i.e. if S is a process
for which there is a sequence of stopping times (Tn )n≥1 such that Tn → ∞ a.s. and
S Tn is a classical semimartingale for all
P n then S is a classical semimartingale. Say
S Tn = M n + An . If one defines M := n≥1 M n 1(Tn−1 ,Tn ] then one can prove that M
P
is a local martingale and A := S − M = n≥1 An 1(Tn−1 ,Tn ] is of finite variation.
3.1.2 Theorem (Bichteler ’79, Dellacherie ’80). For an adapted, càdlàg, real valued process S the following are equivalent.
(i) S is a semimartingale.
(ii) S is a classical semimartingale.
The following, seemingly stronger, theorem is true. This is the theorem that is
stated in the textbook and the one that we will eventually prove.
3.1.3 Theorem. For an adapted, càdlàg, real valued process S the following are
equivalent.
(i) S is a semimartingale.
(ii) S is decomposable (i.e. S = M + A where M is a locally square integrable
martingale and A is of finite variation.)
(iii) For all β > 0 there are M and A such that M is a local martingale with jumps
bounded by β and A is of finite variation and S = M + A.
(iv) S is a classical semimartingale.
43
44
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
So far we have seen that (ii) implies (i) (cf. Corollary 2.3.3) and (iii) implies
(ii) (because a local martingale with bounded jumps is locally bounded, and in
particular is a locally square integrable martingale). The theoretically important,
and difficult, results are (iv) implies (iii) and (i) implies (iv). The former is important enough to be restated as a theorem.
3.1.4 Theorem (Fundamental theorem of local martingales).
If M is a local martingale then for all β > 0 there are N and B such that M = N +B,
where N is a local martingale with jumps bounded by β and B is of finite variation.
Connexion to finance
Suppose that S is a price process, i.e. S t is the price of an asset at time t, where
we make no assumptions about S other than that it is càdlàg. Let H t be the
number of units of S that you hold at time t, so that H is a simple predictable
process. The accumulated gains and losses after following the strategy H up to
Rt
time t are (H · S) t = 0 Hs dSs (the integral makes sense for any S when H is a
simple predictable process). An arbitrage over the time horizon [0, T ] is a strategy
H such that H0 = 0, (H · S) T ≥ 0 a.s., and P[(H · S) T > 0] > 0. It is very important
to be able to determine when there are no possible arbitrages.
3.1.5 Definition. We say that S allows a free lunch with vanishing risk, or FLVR,
for simple integrands if there is a sequence of simple predictable processes (H n )n≥1
such that both of the following conditions hold.
(FL) (H n · S)+
T does not converge to zero in probability.
(VR) limn→∞ k(H n · S)− ku = 0.
Alternatively, S satisfies no free lunch with vanishing risk, or NFLVR, for simple integrands if, for all sequences of simple predictable processes (H n )n≥1 , (VR) implies
(H n · S)+
T → 0 in probability.
It is clear that if S admits an arbitrage then S allows FLVR. In 1994 Delbaen
and Schachermayer proved that NFLVR for simple integrands implies S is a semimartingale. Their proof relies on the Bichteler-Dellacherie theorem.
3.1.6 Definition. We say that S allows free lunch with vanishing risk for simple
integrands with little investment, or FLVR+LI if there is a sequence of simple predictable processes (H n )n≥1 such that (FL) and (VR) hold and so does
(LI) limn→∞ kH n ku = 0.
Alternatively, S satisfies no free lunch with vanishing risk and little investment, or
NFLVR+LI, for simple intgrands if, for all sequences of simple predictable processes
(H n )n≥1 , (VR) and (LI) together imply (H n · S)+
T → 0 in probability.
3.1.7 Theorem. For a locally bounded, adapted, càdlàg process S the following
are equivalent.
(i) S satisfies NFLVR+LI.
(ii) S is a classical semimartingale.
3.2. Proofs of Theorems 3.1.7 and 3.1.8
45
3.1.8 Theorem. For an adapted càdlàg process S the following are equivalent.
(i) For all sequences (H n )n≥1 of simple predictable processes,
a) limn→∞ kH n ku = 0, i.e. (LI)
b) limn→∞ sup0≤t≤T (H n · S)−
t = 0 in probability (this is slightly weaker
than (VR)).
together imply (H n · S)+
T → 0 in probability.
(ii) S is a classical semimartingale.
Remark. The assertions (ii) implies (i) in both Theorems 3.1.7 and 3.1.8 follow
from (iv) implies (i) in Theorem 3.1.3, because (i) in each theorem holds trivially for semimartingales. Also note that (i) implies (iv) in Theorem 3.1.3 is a
consequence of Theorem 3.1.8.
3.2
Proofs of Theorems 3.1.7 and 3.1.8
First we want to show that if S is locally bounded, càdlàg, and adapted and satisfies NFLVR+LI then S is a classical semimartingale.
Note that the collection of processes satisfying NFLVR+LI is stable under stopping. Since the property of being a classical semimartingale is local and S is locally
bounded, we may assume that kSku ≤ 1. Without loss of generality we assume
that S0 = 0 and the time interval is [0, 1].
Notation. Let Dn denote the dyadic rationals inS
[0, 1] with denominator at most
2n , i.e. Dn := { j2−n : j = 0, . . . , 2n }. Let D := n≥1 Dn denote the collection of
dyadic rationals in [0, 1].
For all n ≥ 1 define S n := M n + An , where M n and An are defined on the set
Dn inductively by An0 := 0, M0n = S0 , and for j = 1, . . . , 2n ,
Anj2−n − An( j−1)2−n := E[S j2−n − S( j−1)2−n |F( j−1)2−n ]
n
n
M j2
−n := S j2−n − A j2−n .
(3.1)
(3.2)
Then M n is a martingale and An is predictable and of finite variation. The hope is
that we may pass to the limit as n → ∞ in some sense. We are assuming that S is
locally bounded, so there are two cases.
CASE 1. If (M n )n≥1 and (An )n≥1 remain bounded in some sense then, by using
Komlós’ lemma, one can pass to the limit to obtain a local martingale M and
a predictable process A of finite variation such that S = M + A.
CASE 2. If (M n )n≥1 or (An )n≥1 does not remain bounded then neither does the
other, and one can construct a sequence of strategies witnessing that S allows FLVR+LI. The idea is that if a predictable process can “get close to” a
martingale then this leads to an arbitrage.
3.2.1 Theorem. Let ( f n )n≥1 be a sequence of random variables.
(KOMLÓS’ LEMMA). If ( f n )n≥1 is bounded in L 1 , i.e. supn≥1 k f n k1 < ∞, then there is
a subsequence ( f nk )k≥1 such that 1k ( f n1 + · · · + f nk ) converges a.s. as k → ∞.
46
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
(MAZUR’S LEMMA). If ( f n )n≥1 is bounded in L 2 , i.e. supn≥1 k f n k2 < ∞, then there
exist g n ∈ conv( f n , f n+1 , . . . ) such that (g n )n≥1 converges a.s. and in L 2 .
PROOF (OF MAZUR’S LEMMA): Let H be the Hilbert space generated by the f n ’s. For
all n define Kn = conv( f n , f n+1 , . . . ), where the closure is taken in the L 2 -norm. In
fact, Kn coincides with the weak closure of conv( f n , f n+1 , . . . ) by a theorem from
functional analysis. By the Banach-AlaogluT
theorem we know that Kn is weakly
compact for all n. Since Kn+1 ⊆ Kn for all n, n≥1 Kn 6= ∅ by the finite intersection
property of compact sets. Hence there is a sequence T
of g n ∈ conv( f n , f n+1 , . . . )
such that (g n )n≥1 converges in L 2 to an element of n≥1 Kn . By passing to a
subsequence we may assume that (g n )n≥1 converges a.s.
ƒ
Remark. We can apply Mazur’s lemma to countably many sequences simultaneously. More precisely, suppose that ( f nm )n≥1 is bounded in L 2 for all m. Then
for all n there are convex weights (λnn , . . . , λnNn ), independent of m, such that the
sequences (λnn f nm + · · · + λnNn f nm )n≥1 converge a.s. and in L 2 for all m.
3.2.2 Proposition. Let S = (S t )0≤t≤1 be càdlàg and adapted, with S0 = 0 and
such that kSku ≤ 1 and S satisfies NFLVR+LI. Let An and M n be as defined in (3.1)
and (3.2). For all " > 0 there is C > 0 and a sequence of stopping times (ρn )n≥1
such that, for all n,
(i) ρn takes values in Dn ∪ {∞}.
(ii) P[ρn < ∞] < ".
n,ρ
(iii) The stopped processes An,ρn and M n,ρn satisfy, for all n, kM1 n k2L 2 ≤ C and
n,ρn
TV(A
2n X
An,ρ−nn − An,ρn −n ≤ C.
) :=
j2
( j−1)2
j=1
3.2.3 Lemma. Under the same assumptions as in Proposition 3.2.2, with
Q n :=
2n
X
(S j2−n − S( j−1)2−n )2 ,
j=1
the sequence (Q n )n≥1 is bounded in probability (i.e. for all α > 0 there is M such
that P[Q n ≥ M ] < α for all n).
PROOF: For all n, let H n := −
P2n
j=1 S( j−1)2
−n
1(( j−1)2−n , j2n ] , a simple predictable pro-
cess. Recall that −a(b − a) = 12 (a − b)2 + 12 (a2 − b2 ).
(H n · S) t = −
2n
X
S t∧( j−1)2−n (S t∧ j2−n − S t∧( j−1)2−n )
j=1
n
=
2
1X
1
(S t∧ j2−n − S t∧( j−1)2−n )2 + (S02 − S t2 )
2 j=1
2
3.2. Proofs of Theorems 3.1.7 and 3.1.8
47
1
1
Q n + (S01 − S t2 )
2
2
=
Clearly kH n ku ≤ 1, and (H n · S) t ≥ − 12 for all t because kSku ≤ 1 and Q n is
nonnegative. Assume for contradiction that (Q n )n≥1 is not bounded in L 0 . Then
there is α > 0 such that for all m > 0 there is nm such that P[(H nm · S)1 ≥ m] ≥ α.
Whence (H nm /m)m≥1 would be a FLVR+LI.
ƒ
For c > 0 define a sequence of stopping times
σn (c) := inf
k
2n
:
k
X
(S j2−n − S( j−1)2−n )2 ≥ c − 4 .
j=1
Given " > 0 there is c1 such that P[σn (c1 ) < ∞] < "/2 by Lemma 3.2.3.
3.2.4 Lemma. Under the same assumptions as Proposition 3.2.2, the stopped
n,σ (c )
martingales M n,σn (c1 ) satisfy kM1 n 1 k2L 2 ≤ c1 .
PROOF: For n ≥ 1 and k = 1, . . . , 2n , since the An s are predictable and the M n s are
martingales,
σ (c )
σ (c )
σ (c )
σ (c )
σ (c )
σ (c )
2
2
2
n 1
n 1
n 1
n 1
n 1
E[(Sk2n−n1 − S(k−1)2
− M(k−1)2
− A(k−1)2
−n ) ] = E[(M k2−n
−n ) ] + E[(A k2−n
−n ) ]
σ (c )
σ (c )
2
n 1
≥ E[(Mk2n−n 1 )2 − (M(k−1)2
−n ) ]
σ (c1 ) 2
) ] as a telescoping series and simplify to get
X
σ (c )
σ (c )
σn (c1 )
2
2
E[(M1 n 1 )2 ] =
E[(Sk2n−n1 − S(k−1)2
−n ) ] + E[(Sσn (c1 ) − Sσn (c1 )−2−n ) ]
Write E[(M1 n
k2−n ≤σn (c1 )
≤ (c1 − 4) + 22 = c1 .
ƒ
P2n (σ (c )∧1) n
3.2.5 Lemma. Let V n := TV(An,σn (c1 ) ) = i=1 n 1
|A j2−n − An( j−1)2−n |. Under the
n
assumptions of Proposition 3.2.2 the sequence (V )n≥1 is bounded in probability.
PROOF: Assume for contradiction that (V n )n≥1 is not bounded in probability. Then
there is α > 0 such that for all k there is nk such that P[V nk ≥ k] ≥ α. For n ≥ 1
define
n,σ (c )
n,σn (c1 )
b nj−1 := sign A j2−nn 1 − A( j−1)2
∈ F( j−1)2−n
−n
and H n (t) :=
P2n
j=1
b nj−1 1(( j−1)2n , j2−n ] (t). Then kH n ku ≤ 1 and
(H n,σn (c1 ) · S) t =
X
j≤bt2n c
σ (c )
σn (c1 )
σ (c )
σ (c )
n
b nj2−n S j2n−n 1 − S( j−1)2
+ bbt2
S t n 1 − Sbt2n n c1
nc
−n
≥ (H n,σn (c1 ) · An )bt2n c2−n + (H n,σn (c1 ) · M n )bt2n c2−n − 2
48
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
and at time t = 1 we have (H n,σn (c1 ) · S)1 = V n + (H n,σn (c1 ) · M n )1 . But the second
summand is bounded in L 2 (it is at most c1 by Lemma 3.2.4), so we conclude that
(H n,σn (c1 ) · S)1 is not bounded in probability.
Define a sequence of stopping times
j
n,σn (c1 )
n
ηn (c) := inf
: |(H
· M ) j2−n ≥ c .
2n
Because E[(sup1≤ j≤2n ((hn,σn (c1 ) · M n ) j2−n )2 ] ≤ 4c1 by Doob’s sub-martingale inequality, (H n,σn (c1 ) · M n ) is bounded in probability. Therefore there is c 0 > 0 such
0
that P[ηn (c 0 ) < ∞] ≤ α/2. Note that H n,σn (c1 )∧ηn (c ) · S is (uniformly) bounded
0
n,σn (c1 )∧ηn (c 0 )
below by c . We claim (H
· S)1 is not bounded in probability. Indeed,
for any n and any k,
α ≤ P[(H n,σn (c1 )∧ηn (c ) · S)1 ≥ k]
0
≤ P[(H n,σn (c1 ) · S)1 ≥ k, ηn (c 0 ) = ∞] + P[ηn (c 0 ) < ∞].
Since P[ηn (c 0 ) < ∞] ≤ α/2, the probability of the other event is at least α/2. This
gives the desired contradiction because it is now easy to construct a FLVR+LI. ƒ
PROOF (OF PROPOSITION 3.2.2): Define a sequence of stopping times
τn (c) := inf
k
2n
:
k
X
|Anj2−n − An( j−1)2−n | ≥ c .
j=1
By Lemma 3.2.5 there is c2 such that P[τn (c2 ) < ∞] < "/2. Take C := c1 ∨ c2 and
ρn := σn (c1 ) ∧ τn (c2 ).
ƒ
3.2.6 Lemma. Let f , g : [0, 1] → R be measurable functions, where f is left
PK
continuous and takes finitely many values. Say f = k=1 f (sk )1(sk−1 ,sk ] . Define
( f · g)(t) :=
K
X
f (sk−1 )(g(sk ) − g(sk−1 )) + f (sk(t) )(g(t) − g(sk(t) ))
k=1
where k(t) is the biggest of the k such that sk less than or equal to t. Then for all
partitions 0 ≤ t 0 ≤ · · · ≤ t M ≤ 1,
M
X
i=1
|( f · g)(t i ) − ( f · g)(t i−1 )| ≤ 2 TV( f )kgk∞ +
X
M
|g(t i ) − g(t i−1 )| k f k∞ .
i=1
3.2.7 Proposition. Let S = (S t )0≤t≤1 be càdlàg and adapted, with S0 = 0 and such
that kSku ≤ 1 and S satisfies NFLVR+LI. For all " > 0 there is C and a [0, 1] ∪ {∞}
valued stopping time α such that P[α < ∞] < " and sequences (Mn )n≥1 and
(An )n≥1 of continuous time càdlàg processes such that, for all n,
(i) A0n = M0n = 0
3.2. Proofs of Theorems 3.1.7 and 3.1.8
49
(ii) S α = An,α + Mn,α
n,α
(iii) Mn,α is a martingale with kM1 k2L 2 ≤ C
P2n
n,α
n,α
(iv)
j=1 |A j2−n − A( j−1)2−n | ≤ C.
PROOF: Let " > 0 be given. Let C, M n , An , and ρn be as in Proposition 3.2.2.
Extend M n and An to all t ∈ [0, 1] by defining M tn := E[M1n |F t ] and Ant = S t − M t .
Note that the extended An is no longer necessarily predictable, and currently we
only have control of the total variation of An,ρn over Dn , i.e.
2n (ρ
n ∧1)
X
|Anj2−n − An( j−1)2−n | ≤ C.
j=1
Notice that, for t ∈ (( j − 1)2−n , j2−n ],
Ant = S t − M tn
n
= S t − E[M j2
−n |F t ]
= S t − E[S j2−n − Anj2−n |F t ]
= Anj2−n − (E[S j2−n |F t ] − S t )
From this and kSku ≤ 1 it follows that kAnt − Anj2−n k∞ ≤ 2, so kAn,ρn ku ≤ C + 2.
How do we find the “limit” of sequence of stopping times (ρn )n≥1 ? The trick
is to define Rn := 1[0,ρn ∧1] , a simple predictable process, and note that stopping at
ρn is like integrating Rn , i.e. An,ρn = Rn · An and M n,ρn = Rn · M n . We have that
1 ≥ E[Rn1 ] = E[1ρn =∞ ] = 1 − P[ρn < ∞] ≥ 1 − ".
Apply Komlós’ lemma to (Rn1 )n≥1 to obtain convex weights (µnn , . . . , µnNn ) such that
Rn :=
∞
X
µni Ri1 → R1 a.s. as n → ∞
i=n
By the dominated convergence theorem, E[R1 ] ≥ 1 − ". Observe that
Rn · S =
∞
X
µni (Ri · M i ) +
2
µni (Ri · Ai )
i=n
i=n
|
∞
X
{z
L norm ≤
p
C
}
|
{z
}
TV over Dn is ≤ C
Define αn := inf{t : Rnt ≤ 12 }. Each Rn is a left continuous, decreasing process.
In particular, Rαn n ≥ 12 > 0, so we can divide by this quantity. We claim that
P[αn < ∞] < ". Indeed, on the event [αn < ∞], R1n ≤ Rαn n + ≤
" ≥ E[1 − R1n ] ≥ E[(1 − Rαn n + )1αn <∞ ] ≥
1
2
1
2
so
P[αn < ∞].
50
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
Define new processes Ttn := 1[0,αn ] (t)/Rnt . Then kT n ku ≤ 2 and T n · (Rn · S) = S αn .
Thus we define Mn and An by
S αn = T n ·
X
∞
X
∞
µni (Ri · M i ) + T n ·
µni (Ri · Ai ) =: Mn + An .
i=n
i=n
The total variation of T over Dn is bounded by 3. By Lemma 3.2.6,
n
2n
X
|Anj2−n
− A(nj−1)2−n |
j=1
∞
X
n
i
i ≤ 2 TVn (T )
µi (R · A )
n
∞
i=n
+ kT k∞ TVn
n
X
∞
µni (Ri
·A )
i
i=n
≤ 6(C + 2) + 2C
n,α
That kM1n k2L 2 ≤ C follows from the fact that kM1 n k2L 2 ≤ C. To finish the proof,
we show that there is a subsequence (αnk )k≥1 such that α := infk αnk satisfies
P[α < ∞] ≤ 4". We know P[R1 ≤ 23 ] ≤ 3" because E[R1 ] ≥ 1 − ". Since R1n → R1
a.s. there is a subsequence such that P[|R1n − R1 | ≥
1
]
15
≤ "2−k . Finally,
3
n
P[α < ∞] ≤ P inf R1k ≤
k
5
3
2
n
≤ 3" + P inf R1k ≤ , R1 >
k
5
3
∞
X
3
2
nk
≤ 3" +
P R1 ≤ , R1 >
5
3
k=1
∞
X
1
nk
≤ 3" +
P |R1 − R1 | ≥
15
k=1
≤ 4"
Therefore (Mn )n≥1 , (An )n≥1 , and α have the desired properties.
ƒ
Remark. One thing to take away from this is that if you need to take a “limit of
stopping times” then one way to do it is to turn the stopping times into processes
and take the limits of the processes.
PROOF (OF (I) IMPLIES (II) IN THEOREM 3.1.7):
We may assume the hypothesis of Proposition 3.2.7. Let " > 0 and take C, α,
(Mn )n≥1 , and (An )n≥1 as in Proposition 3.2.7. Apply Komlós’ lemma to find convex
weights (λnn , . . . , λnNn ) such that
n,α
N ,α
λnn M1 + · · · + λnNn M1 n → M1
N ,α
n
n
λnn An,α
→ At
t + · · · + λNn A t
3.2. Proofs of Theorems 3.1.7 and 3.1.8
51
for all t ∈ D, where the convergence is a.s. (and also in L 2 for M). For all n,
2n
X
n,α
n,α
|A j2−n − A( j−1)2−n | ≤ C
j=1
so the total variation of A over D is bounded by C. Further, we have S α = M t + A t
(where M t = E[M1 |F t ]). A is càdlàg on D, so define it on all of [0, 1] to make it
càdlàg. M is an L 2 martingale so it has a càdlàg modification. Since P[α < ∞] < "
and " > 0 was arbitrary, and the class of classical semimartingales is local, S must
be a classical semimartingale.
ƒ
Remark. On the homework there is an example of a (not locally bounded) adapted
càdlàg process that satisfies NFLVR and is not a classical semimartingale.
PROOF (OF (I) IMPLIES (II) IN THEOREM 3.1.8):
We no longer assume that S is locally bounded. The trick is to leverage the result for locally bounded processes by subtracting the “big”Pjumps from S. Assume
without loss of generality that S0 = 0 and define J t := s≤t ∆Ss 1|∆Ss |≥1 . Then
X := S − J is an adapted, càdlàg, locally bounded process. We will show that Theorem 3.1.8(i) for S implies NFLVR+LI for X , so that we may apply Theorem 3.1.7
to X . Then since J is of finite variation, this will then imply S is a classical semimartingale.
Suppose H n ∈ S are such that kH n ku → 0 and k(H n · X )− ku → 0. We need to prove
that (H n · X ) T → 0 in probability. First we will show that k(H n · S)− ku → 0.
n
n
−
sup (H n · S)−
t ≤ sup (H · X ) t + sup |(H · J) t |
0≤t≤T
0≤t≤T
0≤t≤T
n
≤ sup (H n · X )−
t + (kH k∞ · TV(J)) T
0≤t≤T
→ 0 by the assumptions on H n
By (i), (H n · S) T → 0 in probability. Since (H n · J) → 0 in probability (as J is
a semimartingale), we conclude that (H n · X ) T = (H n · S) T − (H n · J) T → 0 in
probability. Therefore X satisfies NFLVR+LI.
ƒ
Remark.
(i) S satisfies NFLVR if k(H n ·S)− ku → 0 implies (H n ·S) T → 0 in probability. This
is equivalent to the statement that {(H · S) T : H ∈ S, H0 = 0, k(H · S)− ku ≤ 1}
is bounded in probability. The latter is sometimes called no unbounded profit
with bounded risk or NUPBR.
(ii) It is important to note that this definition of NFLVR does not imply that there
is no arbitrage. The usual definition of NFLVR in the literature is as follows.
Let K := {(H · S) T : H ∈ S, H0 = 0} and let C := { f ∈ L ∞ : f ≤ g for some
∞
= {0} (the closure
g ∈ K}. It is said that there is NFLVR if and only if C ∩ L+
∞
is taken in the norm topology of L ). This is equivalent to NUPBR+NA,
∞
(where NA is no arbitrage, the condition that K ∩ L+
= {0}).
52
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
3.3
A short proof of the Doob-Meyer theorem
3.3.1 Definition. An adapted process S is said to be of class D if the collection of
random variables {Sτ : τ is a finite valued stopping time} is uniformly integrable.
càdlàg + predictable implies what?
Note that *-martingale means càdlàg.
3.3.2 Theorem (Doob-Meyer for sub-martingales). Let S = (S t )0≤t≤T be a submartingale of class D. Then S can be written uniquely as S = S0 + M + A where M
is a u.i. martingale with M0 = 0 and A is a càdlàg, increasing, predictable process
with A0 = 0.
We will require the following Komlós-type lemma to obtain a limit in the proof
of the theorem.
3.3.3 Lemma. If ( f n )n≥1 is a u.i. sequence of random variables then there are
g n ∈ conv( f n , f n+1 , . . . ) such that (g n )n≥1 converges in L 1 .
The details need to be filled in here.
PROOF: Define f nk = f n 1| f n |≤k for all n, k ∈ N. Then f nk ∈ L 2 for all n and k since
they are bounded random variables. As noted before, there are convex weights
(λnn , . . . , λnNn ) and f k ∈ L 2 such that λnn f nk + · · · + λnNn f Nkn → f k ∈ L 2 . Since ( f n )n≥1
is u.i.,
lim sup k(λnn f nk + · · · + λnNn f Nkn ) − (λnn f n + · · · + λnNn f Nn )k1 = 0
k→∞
n
which implies that (λnn f n + · · · + λnNn f Nn )n≥0 is Cauchy in L 1 .
ƒ
PROOF (OF THE DOOB -MEYER THEOREM FOR SUB -MARTINGALES):
Assume without loss of generality that T = 1. Subtracting the u.i. martingale
(E[S1 |F t ])0≤t≤1 from S we may further assume that S1 = 0 and S t ≤ 0 for all t.
Define An0 = 0 and, for t ∈ Dn ,
Ant − Ant−2−n := E[S t − S t−2−n |F t−2−n ]
M tn := S t − Ant .
An = (Ant ) t∈Dn is predictable and it is increasing since S is a sub-martingale. Furthermore, M1n = −An1 since S1 = 0, and in fact M tn = − E[An1 |F t ] for all t ∈ Dn .
Therefore for every Dn valued stopping time τ,
Sτ = − E[An1 |Fτ ] + Anτ .
We would like to show that (An1 )n≥1 is u.i. (and hence that (M1n )n≥1 is u.i.). For
c > 0 define τn (c) := inf{( j − 1)2−n : Anj2−n > c} ∧ 1, which is a stopping time since
An is predictable. Then Aτn (c) ≤ c and {An1 > c} = {τn (c) < 1}, so
E[An1 1An1 >c ] = E[An1 1τn (c)<1 ]
= E[E[An1 |Fτn (c) ]1τn (c)<1 ]
3.3. A short proof of the Doob-Meyer theorem
= E[(−Sτn (c) + Anτ
n (c)
53
)1τn (c)<1 ]
≤ − E[Sτn (c) 1τn (c)<1 ] + c P[τn (c) < 1]
Also, {τn (c) < 1} ⊆ {τn (c/2) < 1}, so
− E[Sτn (c/2) 1τn (c/2)<1 ] = E[(An1 − Anτ
≥
≥
E[(An1
c
2
)1τn (c/2)<1 ]
n (c/2)
n
− Aτ (c/2) )1τn (c)<1 ]
n
P[τn (c) < 1]
Whence E[An1 1An1 >c ] ≤ −2 E[Sτn (c/2) 1τn (c/2)<1 ] − E[Sτn (c) 1τn (c)<1 ] We may now apply the class D assumption on S to show that (An1 )n≥1 is u.i., provided that we can
show P[τn (c) < 1] → 0 as c → ∞ uniformly in n. But, for any n,
P[τn (c) < 1] = P[An1 > c] ≤
1
1
1
E[An1 ] = − E[M1n ] = − E[S0 ]
c
c
c
which goes to zero as c → ∞. Therefore (An1 )n≥1 , and hence also (M1n )n≥1 , are u.i.
sequences of random variables.
Extend M n to all t ∈ [0, 1] by defining M tn := E[M1n |F t ]. By Lemma 3.3.3 there
are convex weights λnn , . . . , λnNn and M ∈ L 1 such that the sequence of martingales
Mn := λnn M n + · · · + λnNn M Nn satisfies M1n → M in L 1 . Then Mnt → E[M |F t ] =: M t Fill in the details (use Jensen’s inequality?).
as well.
P
n
n
n
Extend A to on [0, 1] by defining it to be t∈Dn A t 1(t−2−n ,t] . Note that A
extended in this way is a predictable process. Define An := λnn An + · · · + λnNn ANn
and A t := S t − M t . For all t ∈ D,
Ant := (S t − Mnt ) → A t in L 1
and, by passing to a subsequence, we may assume the convergence is a.s. This implies in particular that (A t ) t∈D is increasing, so by right continuity, so is (A t ) t∈[0,1] .
Further, An is predictable for all n. If we can prove that lim supn Ant (ω) = A t (ω)
a.s. for all t ∈ [0, 1] then this would imply that A is a predictable process and we
would be done.
By right continuity lim supn Ant ≤ A t for all t and limn→∞ Ant = A t if A is continuous at t. Since A is càdlàg there are countably many jump times. To prove what
we want it suffices to prove, for all stopping times τ, lim supn Aτn = Aτ a.s. Fix a
stopping time τ. Since Aτn ≤ A1n → A1 in L 1 , we may apply Fatou’s Lemma to get
lim inf E[Anτ ] ≤ lim sup E[Aτn ] ≤ E[lim sup Aτn ] ≤ E[Aτ ]
n
n
n
Therefore it is actually enough to show that limn→∞ E[Anτ ] = E[Aτ ]. Let σn :=
inf{t ∈ Dn : t ≥ τ}, a stopping time. Note that σn ↓ τ as n → ∞. By the definition
of An , Anτ = Anσn , so
lim E[Anτ ] = lim E[Anσn ]
n→∞
n→∞
54
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
= lim E[Sσn ] − E[Mσn ]
discrete Doob-Meyer
= E[Sτ ] − E[M0 ]
since S is class D
n→∞
= E[Aτ ].
3.4
ƒ
Fundamental theorem of local martingales
3.4.1 Definition. A stopping time T is predictable there there is a sequence of
stopping times (Tn )n≥1 , called an announcing sequence, such that Tn < T on the
event {T > 0} and Tn ↑ T . A stopping time T is said to be totally inaccessible if
P[T = S < ∞] = 0 for all predictable stopping times S.
3.4.2 Example. Suppose that X is a continuous process. T := inf{t : X t ≥ K} is
predictable and Tn := inf{t : X t ≥ K − 1n } is an announcing sequence. The first
jump time of a Poisson process is totally inaccessible.
3.4.3 Theorem. If S is a càdlàg sub-martingale of class D and S jumps only at
totally inaccessible stopping times then S can be written uniquely as S = S0 +M +A,
where M is a u.i. martingale with M0 = 0, and A is continuous, increasing, and
A0 = 0.
PROOF: This is Theorem 10 in Chapter III of the textbook. The proof of uniqueness
in this case is easy, since if M + A = S = M 0 + A0 are two decompositions then
A − A0 is a continuous martingale of finite variation null at zero. Hence A − A0 ≡ 0
by Theorem 2.6.10. See the textbook for the proof of existence.
ƒ
Remark. A stronger statement is true. If S is as in the statement of the Theorem 3.4.3 and S jumps at a totally inaccessible stopping time T then A in the
decomposition is continuous at T .
3.4.4 Definition. Let A be an adapted process of finite variation with A0 = 0. If A
satisfies any of the following equivalent conditions then A is said to be a natural
process.
(i) E[|A|∞ ] < ∞ (i.e. A is of integrable variation) and E[[M , A]∞ ] = 0 for all
bounded
martingales M .
R∞
(ii) E[ 0 Ms− dAs ] = E[M∞ A∞ ] for all all bounded martingales M . (This is an
integration-by-parts
formula, in some sense.)
R∞
R
∞
(iii) E[ 0 Ms− dAs ] = E[ 0 Ms dAs ] for all all bounded martingales M .
R∞
(iv) E[ 0 ∆Ms− dAs ] = 0 for all all bounded martingales M .
(v) [M , A] is a martingale for all bounded martingales M .
Remark. A càdlàg process locally of integrable variation is of finite variation.
3.4. Fundamental theorem of local martingales
3.4.5 Theorem. If A is a natural process and a martingale then A ≡ 0.
PROOF: Fix a finite valued stopping time T and let H be a bounded, non-negative
martingale. It can be shown that, since A is a martingale,
Z T
Hs− dAs
E
=0
0
using dominated convergence and the fact that it is trivial if A is a simple process (take the limit of the Riemann sums). This implies, since A is natural, that
E[H T A T ] = 0. Now take H t := E[1AT >0 |F t ], a bounded, non-negative martingale.
Then
E[1AT >0 A T ] = E[E[1AT >0 |F T ]A T ] = E[H T A T ] = 0.
Therefore A T ≤ 0. But since E[A T ] = 0, it must be the case that A T = 0. Since this
holds for any finite valued stopping time T , A ≡ 0.
ƒ
3.4.6 Theorem. If A is a predictable process of integrable variation null at zero
then A is a natural process.
PROOF: This is Theorem 14 in the textbook. There a different proof in Jacod and
Shiryaev as well.
ƒ
3.4.7 Lemma. If Y is a u.i. martingale of finite variation then Y is locally of integrable variation.
PROOF: Define τn := inf{t : |Yt | > n}. For any t we have |Yt− | ≤ |Y0 | + |Y | t− and
hence
(∆|Y |) t = |∆Yt | ≤ |Yt | + |Yt− | ≤ |Yt | + |Y0 | + |Y | t− .
Thus
|Y |τn ≤ |Y |τn − + (∆|Y |)τn ≤ 2|Y |τn − + |Y0 | + |Yτn | ≤ 2n + |Y0 | + |Yτn | ∈ L 1 .
Therefore (τn )n≥1 is a localizing sequence for |Y | relative to the class increasing
processes integrable at ∞.
ƒ
3.4.8 Theorem. If M is a predictable local martingale of finite variation then M
is constant.
PROOF: Suppose that (Tn )n≥1 is a localizing sequence for M . If X n := (M − M0 ) Tn
then X n ∈ P , X 0n = 0, X n is u.i., and it is of finite variation. By Lemma 3.4.7,
X is locally of integrable variation. Without loss of generality assume that it is of
integrable variation. By Theorem 3.4.6, X n is natural. By Theorem 3.4.5, X n is
the zero process.
ƒ
3.4.9 Corollary. The Doob-Meyer decomposition is unique.
55
56
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
3.4.10 Theorem (Doob-Meyer without class D).
Suppose that S is a super-martingale. Then S can be decomposed uniquely as
S = S0 + M − A where M is a local martingale with M0 = 0 and A is càdlàg,
increasing, and predictable with A0 = 0.
PROOF: Define T m := inf{t : |S t | ≥ m} ∧ m. By the optional sampling theorem,
m
m
S T m ∈ L 1 and S T is dominated by m ∨ |S T m | ∈ L 1 . Hence S T is of class D,
m
so by the Doob-Meyer decomposition we may write S T = M m − Am where M m
is u.i. and Am is increasing and predictable with Am
0 = 0. By the uniqueness in
m+1
Doob-Meyer for class D, it must be the case that M m = MP
and Am = Am+1 on
∞
m
[0, T ]. The desired processes may be defined as M := m=1 M m 1[Tm−1 ,Tm ) and
P∞
A := m=1 Am 1[Tm−1 ,Tm ) .
ƒ
3.5
Quasimartingales, compensators, and the fundamental
theorem of local martingales
3.5.1 Definition. Let X be a process and Π be a partition for which X t i ∈ L 1
for all t i ∈ Π. Define
P the variation of X along Π by varΠ (X ) := E[C(X , Π)],
where by C(X , Π) := Π |X t i − E[X t i+1 |F t i ]|. We say that X is a quasimartingale if
var(X ) := supΠ varΠ (X ) < ∞ (and hence E[|X t |] < ∞ for all t ∈ [0, ∞)).
3.5.2 Theorem (Rao). X on [0, ∞) is a quasimartingale if and only if X = Y − Z
where Y and Z are nonnegative super-martingales.
PROOF: See the textbook, though the proof therein is not so clear.
ƒ
3.5.3 Theorem. A quasimartingale X has a unique decomposition as X = M + A
where M is a local martingale and A is a predictable process locally of integrable
variation.
PROOF: Existence and uniqueness follow from Theorems 3.4.10 and 3.5.2. Let’s
see that A is locally of integrable variation. Suppose X = Y − Z where Y = M Y −AY
and Z = M Z − AZ by the Doob-Meyer decomposition. Clearly A = AY − AZ , so we
need only show that AY and AZ are locally of integrable variation. Let τm be a
localizing sequence for M Y .
Y,τm
m
E[AY,τ
∞ ] = lim E[A t
t→∞
= lim
t→∞
]
Y,τ
τ
E[M t m ] − E[Yt m ]
τ
= Y0 − lim E[Yt m ]
t→∞
1
≤ Y0 ∈ L
and the same argument works for AZ .
ƒ
3.5. Quasimartingales, compensators, and the fundamental theorem of local martingales
3.5.4 Corollary (Existence of the compensator). Let A be a process locally of
integrable variation. There is a unique finite variation process Ap ∈ P such that
A − Ap is a local martingale.
Remark. Ap is called the compensator of A. When A = [X , Y ] then Ap =: 〈X , Y 〉,
the predictable variation between X and Y .
PROOF: A is locally a quasimartingale because it is locally of finite variation.
ƒ
3.5.5 Examples.
p
(i) If N is a Poisson of process of rate λ then Nt = λt. We know what N − [N ]
is a local martingale, but in this case [N ] = N . What is meaningful is that
([N ] t − λt) t≥0 is a local martingale.
(ii) Let A t := 1 t≥τ = 1[τ,∞) (t), where τ is a random time. Let F be the minimal
filtration that satisfies the usual conditions and makes τ a stopping time. (F
may be realized as follows. Let F t0 := σ({τ ≤ s} : s ≤ t) ∨ N and take
R t∧τ d F (u)
T
p
0
F t := ">0 F t+"
.) The F-compensator of A is A t = 0 1−F (u−) , where
F is the cumulative distribution function of τ on [0, ∞]. Note that if F is
continuous then so is Ap .
3.5.6 Theorem (Existence of the predictable projection). Let X be a bounded
process that is F ⊗ B measurable. There exists a unique process pX such that
(i) pX ∈ P
(ii) ( pX ) T = E[X T |F T − ] on {T < ∞} for all predictable stopping times T .
3.5.7 Theorem.
(i) If X is a local martingale then pX = X − and p(∆X ) = 0.
(ii) If A is locally of integrable variation and càdlàg then p(∆A) = ∆Ap .
PROOF (OF (II)): By the definition of the compensator, A− Ap is a local martingale.
By (i), p(∆(A − Ap )) = 0, so p(∆A) = p(∆Ap ). Since Ap is itself predictable, so is
∆Ap , and hence p(∆Ap ) = ∆Ap .
ƒ
The following proof is based on the presentation in Jacod and Shiryaev. See
the textbook for a different presentation.
PROOF (OF THE FUNDAMENTAL THEOREM OF LOCAL MARTINGALES):
Assume without loss of generality that M0 = 0. By localization we may assume
that M is a u.i. martingale (thisPrequires work, but we have done it previously).
Let b = β/2 and define A t := s≤t ∆Ms 1|∆Ms |≥b . The process A is well-defined
and of finite variation since M is càdlàg. Let Tn := inf{t : |A| t > n or |M t | > n} so
that |A| Tn = |A| Tn − + |∆A Tn | ≤ n + |∆A Tn | and |∆A Tn | ≤ |∆M Tn | ≤ n + |M Tn |. Hence
|A| Tn ≤ 2n + |M Tn | ∈ L 1 because M is u.i., so therefore |A| is locally integrable and
Ap exists. Let B := A − Ap and N := M − B. Clearly B is of finite variation, and
it is a local martingale by definition of Ap , and N is a local martingale because
57
58
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
it is a difference of local martingales. Define X t := ∆M t 1|∆M t |<b , so that ∆A =
∆M − X . Then p(∆A) = p(∆M ) − pX = − pX by (i) of Theorem 3.5.7. Therefore
∆N = ∆M − ∆A + ∆Ap = X − pX , so |∆N | ≤ |X | + | pX | ≤ 2b, i.e. the jumps of N
are bounded by 2b = β.
ƒ
3.5.8 Corollary.
(i) A classical semimartingale is a semimartingale.
(ii) A càdlàg sub- (super-) martingale is a semimartingale.
PROOF: Cf. Theorems 3.1.4, 2.3.1, 2.3.2 and the Doob-Meyer decomposition.
ƒ
3.5.9 Theorem. If H ∈ L and M is a local martingale then H · M is well-defined
and is a local martingale.
Remark. This theorem is not true if H ∈
/ L. (Example?) This is an important
difference between the general theory of stochastic integration and the theory of
stochastic integration with respect to continuous local martingales. In the latter
the integral of any predictable process is a local martingale.
PROOF: Write M = N + B as in the fundamental theorem of local martingales. By
localization we may assume that H is bounded, M is u.i. and has bounded jumps,
and that B has integrable variation (cf. lemma in the proof of Theorem 3.4.8). We
know by Theorem 3.4.8 that H · N is a locally square integrable martingale. Using
the dominated convergence theorem one can prove that H · B is a martingale (use
“Riemann sums”).
ƒ
3.6
Special semimartingales and another decomposition
theorem for local martingales
3.6.1 Definition. A semimartingale X is said to be a special semimartingale if it
can be written X = X 0 + M + A where M0 = A0 = 0, M is a local martingale, and
A is predictable and of finite variation.
3.6.2 Examples.
(i) Quasimartingales (including sub- and super-martingales).
(ii) Local martingales
(iii) Continuous semimartingales (proved below).
3.6.3 Theorem. If X is a special semimartingale then the decomposition in the
definition is unique. This decomposition is known as the canonical decomposition
of X .
3.6.4 Theorem. The following are equivalent for a semimartingale X .
(i) X is a special semimartingale.
3.6. Special semimartingales and another decomposition theorem for local martingales
(ii) X ∗ , the running maximum of X , is locally integrable.
(iii) J := (∆X )∗ is locally integrable.
(iv) Decomposing X as X = M +A, where M is a local martingale and A has finite
variation, implies that A is locally of integrable variation.
(v) There exists a local martingale M and A locally of integrable variation such
that X = M + A.
PROOF:
(I) IMPLIES (II) Recall the following fact. If A is càdlàg, predictable, and of finite
variation then A is locally of integrable variation. (This is an exercise on the
homework.) Recall also that if M is a local martingale then M t∗ = sups≤t |Ms |
is locally integrable. (Indeed, we may assume that M is u.i. Let Tn := inf{t :
M t∗ > n} and note that (M ∗ ) Tn ≤ n ∨ |M Tn |, which is integrable.) If X is a
special semimartingale then write X = M + A as in the definition and note
that X ∗ ≤ M ∗ + A∗ ≤ M ∗ + |A|, so X t∗ is locally integrable for all t.
(II) IMPLIES (III) For all s ≤ t, |∆X s | = |X s − X s− | ≤ 2X t∗ , which implies J ≤ 2X ∗ .
(III) IMPLIES (IV) Homework.
(IV) IMPLIES (V) Trivial.
(V) IMPLIES (I) Write X = M + A where A is locally of integrable variation. Then
by Corollary 3.5.4, Ap exists and X = (M + A − Ap ) + Ap . Note that the first
summand is a local martingale and the second, Ap , is predictable and of
finite variation.
ƒ
3.6.5 Theorem. If X is a continuous semimartingale then X may be written X =
M + A, where M is a local martingale and A is continuous and of finite variation.
In particular, X is a special semimartingale.
PROOF: By Theorem 3.6.4(iii) X is special, so write X = M + A in the unique way.
Let’s see that A is continuous. We have A ∈ P , so pA = A and p(∆A) = ∆A. But
∆A = p(∆A) = p(∆X ) − p(∆M )
= p(∆X )
p
(∆M ) = 0 by Theorem 3.5.7
=0
X is continuous by assumption.
ƒ
3.6.6 Definition. Two local martingales M and N are orthogonal if M N is a local martingale. In this case we write M ⊥ N . A local martingale M is purely
discontinuous if M0 = 0 and M ⊥ N for all continuous local martingales N .
Remark. If N is a Poisson process of rate λ then Ñt := Nt − λt is purely discontinuous. However, Ñ is not the sum of its jumps, i.e. not locally constant pathwise.
3.6.7 Theorem. Let H2 be the space of L 2 martingales.
(i) For all M ∈ H2 , E[[M ]∞ ] < ∞ and 〈M 〉 exists.
59
60
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
(ii) (M , N )H2 := E[〈M , N 〉∞ ] + E[M0 N0 ] defines an inner product on H2 , and
H2 is a Hilbert space with this inner product.
(iii) M ⊥ N if and only if 〈M , N 〉 ≡ 0, which happens if and only if M T ⊥ N − N0
for all stopping times T .
(iv) Take H2,c to be the collection of martingales in H2 with continuous paths
and H2,d to be the collection of purely discontinuous martingales in H2 .
Then H2,c is closed in H2 and (H2,c )⊥ = H2,d . In particular, H2 = H2,c ⊕
H2,d .
3.6.8 Theorem. Any local martingale M has a unique decomposition M = M0 +
M c + M d , where M c is a continuous local martingale, M d is a purely discontinuous
local martingale, and M0c = M0d = 0.
PROOF (SKETCH): Uniqueness follows from the fact that a continuous local martingale orthogonal to itself must be constant. For existence, write M = M0 + M 0 + M 00
as in the fundamental theorem of local martingales, where M 0 has bounded jumps
and M 00 is of finite variation. Then M 00 is purely discontinuous and M 0 is locally
in H2 . But then M 0 = M c + N where M c is a continuous local martingale and
N ∈ H2,d is purely discontinuous. Hence we may take M d = M 00 + N .
ƒ
Suppose that h : R → R and h(x) = x in a neighbourhood (−b, b) of zero, e.g.
h(x) = x1|x|<1 . Let X be a semimartingale and notice that ∆X s − h(∆X s ) 6= 0 only
if |∆X s | ≥ b. Define
X
∆X s − h(∆X s ).
X̌ (h) t :=
0<s≤t
Then X (h) := X − X 0 − X̌ (h) has bounded jumps, so it is a special semimartingale
by Theorem 3.6.4. Write X (h) = M (h) + B(h) as in that theorem, where M (h) is a
local martingale and B(h) is predictable and of finite variation.
3.6.9 Definition. We call (B, C, ν) the semimartingale characteristics of X associated with h, where
(i) B := B(h)
(ii) C := 〈M (h)c 〉 = [M (h)c ], cf. Theorem 3.6.8, and note that C doesn’t depend
on h by the uniqueness part of that theorem.
(iii) ν is the “compensator” of the random measure µX associated with the jumps
of X .
X
µX (d t, d x) :=
1∆X s 6=0 δ(s,∆X s ) (d t, d x)
s≤t
(Inspired by the Lévy-Khintchine formula and Lévy measures.)
Remark. M (h)c and C do not depend on the function h. We write X c := M (h)c for
the “continuous part” of the semimartingale X . (This does not mean that X − X c
is locally constant.)
3.7. Girsanov’s theorem
61
3.6.10 Example. If X is the Lévy process that appears in the Lévy-Khintchine formula and h(x) = x1|x|<1 then B(h) = αt (α depends on h), C(h) = σ2 t (σ2 does
not depend on h), and ν(d t, d x) = ν(d x) d t, where ν is the Lévy measure (which
depends on h).
3.7
Girsanov’s theorem
Working on the usual setup (Ω, F , F = (F t ) t≥0 , P), we know that if Q P and
X is a P local martingale then X is a Q semimartingale (cf. Theorem 2.2.1). By
the Bichteler-Dellacherie theorem X = M + A = N + B where A and B have finite
variation, M is a P local martingale, and N is a Q local martingale. The question
is whether, knowing M and A, we can describe N and B.
3.7.1 Theorem. If Q |Ft P |Ft for all t (we write Q loc P in this case) and M is
an adapted càdlàg process then the following hold.
(i) There exists a unique Z ≥ 0 such that Z t =
d Q |F t
d P |F t
for all t.
(ii) M Z is a P martingale if and only if M is a Q martingale.
(iii) If M Z is a P local martingale then M is a Q local martingale.
(iv) If M is a Q local martingale with localizing sequence (Tn )n≥1 such that
P[limn→∞ Tn = ∞] = 1 then M Z is a P local martingale.
Remark. A property being “P-local” subtly depends on the measure P via the requirement that the localizing sequence of stopping times (Tn )n≥1 satisfies Tn ↑ ∞
P-a.s. If Q ∼loc P then a property holds P-locally if and only if it holds Q-locally.
3.7.2 Corollary. If Q ∼loc P then M Z is a P local martingale if and only if M is a
Q local martingale.
3.7.3 Theorem (Girsanov-Meyer). Suppose thatRQ ∼loc P and let Z be as in Theorem 3.7.1. If M is a P local martingale then M − Z1 d[Z, M ] is a Q local martingale.
PROOF: By integration-by-parts, Z M −[Z, M ] = Z− · M + M− · Z. By Theorem 3.5.9,
Z M − [Z, M ] is a P local martingale. By Corollary 3.7.2 M − Z1 [Z, M ] is a Q local
martingale. By integration-by-parts,
1
Z
[Z, M ] =
1
Z−
· [Z, M ] + [Z, M ]− ·
1
Z
+ [ Z1 , [Z, M ]]
By Theorem 3.5.9 the middle term is a Q local martingale. We also have
1
Z−
· [Z, M ] + [ Z1 , [Z, M ]] =
1
Z−
· [Z, M ] +
X
1
1
∆ ∆[Z, M ] = · [Z, M ]
Z
Z
62
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
where this last integral is a Lebesgue-Stieltjes integral, since [Z, M ] has finite
variation. Thus Z1 [Z, M ] is a Q local martingale plus Z1 · [Z, M ]. Finally,
M−
1
1
1
· [Z, M ] = M − [Z, M ] + [Z, M ] − · [Z, M ]
Z
Z{z
} |Z
{zZ
}
|
1
Q local mtg
Q local mtg
ƒ
Remark. M = (M − Z1 · [Z, M ]) + Z1 · [Z, M ] is a decomposition showing that M
is a Q-semimartingale. If M is a special semimartingale then this might not be
the canonical decomposition because it may not be the case that Z1 · [Z, M ] is
predictable.
3.7.4 Theorem (Girsanov’s theorem, predictable version).
Suppose that Q loc P and let Z be as in Theorem 3.7.1. If M is a P local martingale and [M , Z] is P locally of integrable variation and 〈M , Z〉 is the P-compensator
of [M , Z] then M 0 = M − Z1 · 〈Z, M 〉 is Q-a.s. well-defined and is a Q local martingale. Moreover, [M c ] is a version of [(M 0 )c ].
PROOF: Let Tn := inf{t : Z t < 1n }. Then A = Z1 〈Z, M 〉 is well-defined on [0, Tn ].
−
Let T = limn→∞ Tn .
\
Q[T < ∞] ≤ Q[ {Tn < ∞}]
n
= lim Q[Tn < ∞]
n→∞
= lim EP [Z Tn 1 Tn <∞ ]
n→∞
≤ lim
n→∞
1
n
=0
(Below all of the arguments are on [0, Tn ].) By integration-by-parts
M Z = M− · Z + Z− · M + [Z, M ]
= M− · Z + Z− · M + [Z, M ] − 〈Z, M 〉 +〈Z, M 〉
{z
}
|
{z
} |
P local mtg (Thm 3.5.9)
P local mtg
Hence Z M − 〈Z, M 〉 is a P-local martingale. Now, A is predictable and of finite
variation.
AZ = A− · Z + Z− · A + [Z, A] = A · Z + Z− · A
(Note that A · Z is well-defined and, by results we will see in Chapter 4, a P local
martingale since A is locally bounded). But Z− · A = 〈Z, M 〉, so AZ − 〈Z, M 〉 is a
P local martingale. Hence Z M − ZA = Z M − 〈Z, M 〉 − (ZA − 〈Z, M 〉) is a P local
martingale. By Theorem 3.7.1 M 0 = M − A is a Q local martingale.
To see that [M c ] is a version of [(M 0 )c ] on uses the fact that, for a local martingale P
N , [N c ] = [N ]c , which is to be proved as an exercise. (Assume that
[N d ] = (∆N )2 when N0 = 0.)
ƒ
3.7. Girsanov’s theorem
63
Remark. If Q loc P and X is a P-semimartingale then X is a Q-semimartingale.
Indeed, if X = M + A is a decomposition under P then M = M 0 + M 00 where
M 0 has bounded jumps and M 00 is of finite variation, so X is M 0 plus a finite
variation process. You can prove that [Z, M 0 ] is locally of integrable variation. By
the previous theorem, X is M 0 − Z1 〈Z, M 0 〉 plus a finite variation process, so is a
−
Q-semimartingale.
dQ
3.7.5 Theorem. Suppose that Q loc P and let Z be the density process d P . Let
R := inf{t : Z t = 0, Z t− = 0}, X be a P-local martingale, and U t := ∆X R 1 t≥R . Then
Rt
p
X t − 0 Z1 d[X , Z]s + U t is a Q-local martingale.
s
How to use Girsanov’s Theorem
Suppose that X is a P-semimartingale and X = X 0 + M + A is a decomposition
with A = J · 〈M 〉 where J ∈ L (there may or may not be such a decomposition).
dQ
Define d P = E (−J · M ) − Z, where d Z = −J Z− d M . If the density process is a
Q-martingale then
Z
1
d〈Z, M 〉
is a Q-local martingale
M−
Z−
d[Z, M ] = −J Z− d[M ]
d〈Z, M 〉 = −J Z− d〈M 〉
Z
M+
J d〈M 〉 = X − X 0
is a Q-local martingale
3.7.6 Theorem. Let M be a local martingale
(KAZAMAKI’S CRITERION). Let S be the collection of bounded stopping times. If M
is continuous and
sup E[exp( 12 M T )] < ∞
T ∈S
then E (M ) is a u.i. martingale.
(NOVIKOV ’S CRITERION). If M is continuous and E[exp( 21 [M ]∞ )] < ∞ then E (M )
is a u.i. martingale.
(LÉPINGLE-MÉMIN). If ∆M > −1 and
X
((1 + ∆Ms ) log(1 + ∆Ms ) − ∆Ms )
A t := 21 〈M c 〉 +
s≤t
has a compensator A such that E [exp(Ap∞ )] < ∞ then E (M ) is a u.i. martingale.
(PROTTER-SHIMBO). If M is locally square integrable such that ∆M > −1 and
p
E[exp( 21 〈M c 〉∞ + 〈M d 〉∞ )] < ∞
then E (M ) is a u.i. martingale.
64
The Bichteler-Dellacherie Theorem and its connexions to arbitrage
3.7.7 Exercise. Why must the supremum in Kazamaki’s criterion be taken over
all bounded stopping times and not just deterministic times?
Chapter 4
General stochastic integration
We know how to integrate elements of L with respect to semimartingales. However, this set of integrands is not rich enough to prove “martingale representation”
theorems nor many formulae regarding semimartingale local time.
4.1
Stochastic integrals with respect to predictable processes
For this section we will assume that all semimartingales that appear are null at
zero.
4.1.1 Definition. Let X be a special semimartingale with canonical decomposition
2
2
2
X = M + A. The H2 -norm of X is kX kH2 := k[M ]1/2
∞ k L + k|A|∞ k L . Define H :=
{X : X is a special semimartingale with kX kH2 < ∞}
Remark. This extends the definition given in the previous chapter. A local martingale M is in H2 if and only if M is an L 2 -martingale.
4.1.2 Theorem. (H2 , k · kH2 ) is a Banach space.
n
PROOF (SKETCH): If X n = M n + An is Cauchy in H2 then M∞
is Cauchy in L 2 and
so the sequence has a limit M . As an exercise, read the rest of this proof in the
textbook.
ƒ
4.1.3 Lemma. If H ∈ L and H is bounded and X ∈ H2 then H · X ∈ H2 .
PROOF: Suppose that X = M +A is the canonical decomposition. By Theorem 3.5.9,
H · M is a local martingale. To see that H · A is predictable, write it as a limit of
Riemann sums. It is clearly of finite variation. Also notice
kH · X kH2 ≤ kHku kX kH2 < ∞.
65
ƒ
66
General stochastic integration
It can be seen that
1/2
kH · X kH2 = k(H 2 · [M ])∞
k L 2 + k|H · A|∞ k L 2 .
The integrals on the right hand side are well-defined for any bounded process.
Notation. If F is a σ-algebra then bF denotes the collection of bounded F measurable functions. Let bL denote the collection of bounded processes in L.
4.1.4 Definition. Let X = M + A ∈ H2 and H, J ∈ bP . Define
1/2
dX (H, J) := k((H − J)2 · [M ])∞
k L 2 + k|(H − J) · A|∞ k L 2 .
Remark. dX is not a metric, but it is a semi-metric.
4.1.5 Theorem. If X ∈ H2 then
(i) bL is dense in (bP , dX )
(ii) If (H n )n≥1 ⊆ bL is Cauchy in (bP , dX ) then (H n · X ) is Cauchy in H2 .
(iii) If dX (H n , H) → 0 and dX (J n , H) → 0 with H n , J n ∈ bL and H ∈ bP then
limn→∞ H n · X = limn→∞ J n · X in H2 .
Remark. Theorem 4.1.5(ii) and (iii) can be more succinctly stated as follows. The
mapping (bL, dX ) → (H2 , k · kH2 ) that sends H 7→ H · X is an isometry and can be
extended to (bP , dX ).
PROOF (OF (I)): Let M := bL and
H := {H ∈ bP : for all " > 0 there is J ∈ bL such that dX (H, J) < "}.
Then M is a multiplicative class and H is a monotone vector space. Indeed, H
is a vector space that contains constants (exercise). Suppose that (H n ) ⊆ H and
H n ↑ H pointwise and H is bounded. By the dominated convergence theorem
dX (H n , H) → 0. Let " > 0 and pick n0 such that dX (H n0 , H) < "/2 and pick J ∈ bL
such that dX (H n0 , J) < "/2. Then dX (H, J) < ", so H ∈ H. Therefore H is a
monotone vector space. By the monotone class theorem bP = bσ(M) ⊆ H.
ƒ
4.1.6 Definition. Let X ∈ H2 and H ∈ bP . Suppose that (H n )n≥1 ⊆ bL is such
that dX (H n , H) → 0 (there is such a sequence by Theorem 4.1.5(i)). Define the
stochastic integral of H with respect to X to be
H · X := H2 -lim(H n · X ),
n→∞
The limit exists by Theorem 4.1.5(ii) and H ·X is well-defined (i.e. does not depend
on the choice of sequence) by Theorem 4.1.5(iii).
∗ 2
4.1.7 Theorem. If X ∈ H2 then E[(X ∞
) ] = E[sup t≥0 |X t |2 ] ≤ 8kX k2H2 .
4.1. Stochastic integrals with respect to predictable processes
67
∗
∗
PROOF: Suppose that X = M + A. Then X ∞
≤ M∞
+ |A|∞ . By Doob’s maximal
∗ 2
2
inequality, E[(M∞ ) ] ≤ 4 E[M∞ ] = 4 E[[M ]∞ ]. Whence
∗ 2
∗ 2
(X ∞
) ≤ 2(M∞
) + 2(|A|∞ )2
∗ 2
∗ 2
E[(X ∞
) ] ≤ 2 E[(M∞
) ] + 2 E[(|A|∞ )2 ]
≤ 8 E[[M ]∞ ] + 2 E[(|A|∞ )2 ]
≤ 8kX k2H2
ƒ
4.1.8 Corollary. If X n → X in H2 then there is a subsequence (X nk )k≥1 such that
(X nk − X )∗∞ → 0 a.s.
4.1.9 Theorem (Properties of the stochastic integral).
Let X , Y ∈ H2 and H, K ∈ bP .
(i) (H + K) · X = H · X + K · X
(ii) H · (X + Y ) = H · X + H · Y
(iii) If T is a stopping time then (H · X ) T = (H1[0,T ] ) · X = H · (X T ).
(iv) ∆(H · X ) = H∆X
(v) If T is a stopping time then H · (X T − ) = (H · X ) T − .
(vi) If X has finite variation then H · X coincides with the Lebesgue-Stieltjes integral.
(vii) H · (K · X ) = (H K) · X
(viii) If X is a local martingale then H · X is an L 2 -martingale.
(ix) [H · X , K · Y ] = (H K) · [X , Y ]
4.1.10 Exercise. For (ii), show that you can take the same approximating sequence for dX and dY .
PROOF (SKETCH): For (iv), suppose that (H n )n≥1 ⊆ bL are such that dX (H n , H) →
0 and (H n ·X −H·X )∗∞ → 0 a.s. Then limn→∞ H n ∆X = limn→∞ ∆(H n ·X ) = ∆(H·X ).
It follows that
∆(H · X )
1∆X 6=0 a.s.
lim H n 1∆X 6=0 =
n→∞
∆X
Show that this implies that limn→∞ H n ∆X = H∆X .
For (ix), it is enough to show that [H · X , Y ] = H · [X , Y ]. Again suppose
(H n )n≥1 ⊆ bL are such that dX (H n , H) → 0 and (H n · X − H · X )∗∞ → 0 a.s. Since
Y− is locally bounded we can assume that it is bounded.
[H n · X , Y ] = H n · [X , Y ] → H · [X , Y ]
[H n · X , Y ] = (H n · X )Y − (H n · X )− · Y − Y− · (H n · X )
→ (H · X )Y − (H · X )− · Y − Y− · (H · X )
(This proof may require some further explanation.)
ƒ
68
General stochastic integration
4.1.11 Definition. A property (P) holds prelocally for X if there exist a sequence
(Tn )n≥1 of stopping times with Tn ↑ ∞ such that (P) holds for X Tn − = X 1[0,Tn ) +
X Tn − 1[Tn ,∞) (with the convention that ∞− = ∞).
4.1.12 Theorem. Let X be a semimartingale with X 0 = 0. Then X is prelocally an
element of H2 .
PROOF: Note that X is not necessarily as special semimartingale. Write X = M +
A where M is a local martingale and A is of finite variation (cf. the BichtelerDellacherie Theorem). Let Tn := inf{t : |M t | > n or |A| t > n} and define Y = X Tn − .
Yt∗ = (X Tn − )∗t ≤ (M Tn − )∗t + (ATn − )∗t ≤ 2n
It follows that Y is a special semimartingale. We may assume, by further localization, that [Y ] is bounded. (Indeed, take R n = inf{t : [Y ] t > n} and note that
[Y Rn ]∗ ≤ n+(∆YRn )∗ ≤ 3n.) Suppose that Y = N +B where N is a local martingale
and B is predictable. We have seen that |B| is locally bounded (homework), so we
may assume that B ∈ H2 by localizing once more. Again from the homework,
E[[Y ]∞ ] = E[[N ]∞ ] + E[[B]∞ ], so E[[N ]∞ ] < ∞ and hence N ∈ H2 as well. ƒ
4.1.13 Definition. Let X ∈ H2 with canonical decomposition X = M + A. We say
that H ∈ P is (H2 , X )-integrable if
Z ∞
Z ∞
2 Hs2 d[M ]s + E
E
0
|Hs |d|A|s
< ∞.
0
4.1.14 Theorem. If X ∈ H2 and H ∈ P is (H2 , X )-integrable then the sequence
((H1|H|≤n ) · X )n≥1 is a Cauchy sequence in H2 .
PROOF: Let H n := H1|H|≤n . Clearly H n ∈ bP , so H n · X is well-defined.
kH n · X − H m · X kH2 = dX (H n , H m )
= k((H n − H m )2 · [M ])1/2 k L 2 + k|(H n − H m )2 · A|∞ k L 2
→ 0 by the dominated convergence theorem.
ƒ
4.1.15 Definition. If X ∈ H2 and H ∈ P is (H2 , X )-integrable then define H · X
to be the limit in H2 of ((H1|H|≤n ) · X )n≥1 , which exists by Theorem 4.1.14. Yet
otherwise said,
H · X := H2 -lim(H1|H|≤n ) · X .
n→∞
For any semimartingale X and H ∈ P , we say that H ·X exists if there is a sequence
of stopping times (Tn )n≥1 that witnesses X is prelocally in H2 and such that H is
(H2 , X Tn − )-integrable for all n. In this case we define H · X to be H · (X Tn − ) on
[0, Tn ) and we say that H ∈ L(X ), the set of X -integrable processes.
4.1. Stochastic integrals with respect to predictable processes
69
Remark.
(i) H · X is well-defined because H · (X Tm − ) = (H · X Tn − ) Tm − for n > m (cf. Theorem 4.1.9). By similar reasoning, H · X does not depend on the prelocalizing
sequence chosen.
(ii) If H ∈ (bP )loc then H ∈ L(X ) for any semimartingale X .
(iii) Notice that if H ∈ L(X ) then H · X is a semimartingale. (Prove as an exercise
that if C is the collection of semimartingales then C = Cloc = Cpreloc .)
(iv) Warning: We have not defined H · X = H · M + H · A when X = M + A. If it Is it true?
is true we will need to prove it as a theorem.
4.1.16 Theorem. Let X and Y be semimartingales.
(i) L(X ) is a vector space and Λ : L(X ) → {special semimartingales} : H 7→ H ·X
is a linear map.
(ii) ∆(H · X ) = H∆X for H ∈ L(X ).
(iii) (H · X ) T = (H1[0,T ] ) · X = H · (X T ) for H ∈ L(X ) and stopping times T .
Rt
(iv) If X is of finite variation and 0 |Hs |d|X |s < ∞ then H · X coincides with the
Lebesgue-Stieltjes integral for all H ∈ L(X ).
(v) Let H, K ∈ P . If K ∈ L(X ) then H ∈ L(K · X ) if and only if H K ∈ L(X ), and
in this case (H K) · X = H · (K · X ).
(vi) [H · X , K · Y ] = (H K) · [X , Y ] for H ∈ L(X ) and K ∈ L(Y ).
(vii) X ∈ H2 if and only if supkHku ≤1 k(H · X )∗∞ k L 2 < ∞. In this case,
2
2
kX kH2 ≤ sup k(H · X )∗∞ k L 2 + 2k[X ]1/2
∞ k L ≤ 5kX kH
kHku ≤1
and
kX kH2 ≤ 3 sup k(H · X )∗∞ k L 2 ≤ 9kX kH2 .
kHku ≤1
PROOF (OF (VII)): To be added.
ƒ
4.1.17 Exercise. Suppose that H ∈ P and X is of finite variation and also that
Rt
|Hs |d|X |s < ∞ a.s. for all t (this is the Lebesgue-Stieltjes integral). Is it true
0
that H ∈ L(X )? Note that this is not the same statement as Theorem 4.1.16(iv).
4.1.18 Theorem. Suppose that X is a semimartingale and H ∈ L(X ) under P. If
Q P then H ∈ L(X ) under Q and H ·P X = H ·Q X .
PROOF (SKETCH): Let (Tn )n≥1 witness that H ∈ L(X ) under P, so that Tn ↑ ∞ P-a.s.
dQ
and H is (H2 , X Tn − )-integrable for all n. Let Z t := EP [ d P |F t ], Sn := inf{t : |Z t | >
n} and R n := Tn ∧ Sn . Since |Z Rn − | ≤ n one can show that
1/2
kX kH2 (Q) ≤ sup k(H · X )∗∞ k L 2 (Q) + 2 EQ [[X Rn − ]∞
]
kHku ≤1
= sup EP [ZRn ((H · X )∗∞ )2 ]1/2 + 2 EP [ZRn [X Rn − ]1/2
∞ ]
kHku ≤1
70
General stochastic integration
p
≤ 5 nkX kH2 (P)
(The last inequality is an exercise. There was a hint but I missed it.) This implies
that X ∈ H2 (Q). The proof that H is (H2 , X Rn − )-integrable under Q is similar. ƒ
4.1.19 Theorem. Let M be a local martingale and H ∈ P be locally bounded.
Then H ∈ L(M ) and H · M is a local martingale.
PROOF: We need the following lemma.
4.1.20 Lemma. Suppose that X = M + A, where M ∈ H2 and A is of integrable
variation. If H ∈ P then there is a sequence (H n )n≥1 ∈ bL such that H n ·M → H ·M
in H2 and E[|H n − H| · |A|∞ ] → 0.
PROOF: Let δ̃X (H, J) := E[(H − J)2 · [M ]∞ ]1/2 + E[|H − J| · |A|∞ ] for all H, J ∈ P .
Let H = {H ∈ P : for all " > 0 there is J ∈ bL such that δ̃X (H, J) < "}. One can
see that H is a monotone vector space that contains bL. It follows that bP ⊆ H.ƒ
By localization we may assume that H ∈ bP and M = M 0 + A, where M 0 ∈ H2
and A is of integrable variation (this follows from the fundamental theorem of local
martingales and a problem from the exam). By the lemma there is (H n )n≥1 ∈ bL
such that δ̃X (H n , H) → 0. We know that H n · M = H n · M 0 + H n · A is a martingale
for each n. Finally, H n · M 0 → H · M 0 in H2 and E[|H n − H| · |A|∞ ] → 0 imply that
H · M 0 and H · A are martingales.
ƒ
4.1.21 Theorem (Ansel-Stricker). Let M be a local martingale and H ∈ L(M ).
Then H · M is a local martingale if and only if there stopping times Tn ↑ ∞ and
T
(θn )n≥1 nonpositive integrable random variables such that (H∆M ) t n ≥ θn for all
t and all n.
Remark. If H is locally bounded and Tn is such that M T∗n is integrable and H Tn is
bounded by n then one can take θn := −2nM T∗n in the theorem to show that H · M
is a local martingale.
PROOF: If H · M is a local martingale then there is are Tn ↑ ∞ such that (H · M )∗Tn
is integrable, and one may take θn := −2(H · M )∗Tn to see that the condition is
necessary. Proof of sufficiency may be found in the textbook.
ƒ
4.1.22 Theorem. Let X = M + A be a special semimartingale and H ∈ L(X ). If
H · X is a special semimartingale then H ∈ L(M )∩ L(A), H · M is a local martingale,
and H · A is predictable and of finite variation. In this case H · X = H · M + H · A is
the canonical decomposition of H · X .
4.1.23 Theorem (Dominated convergence). Let X be a semimartingale, G ∈
L(X ), H m ∈ P such that |H m | ≤ G, and assume that H n → H pointwise. Then
H m , H ∈ L(X ) for all m and H n · X → H · X u.c.p.
4.1. Stochastic integrals with respect to predictable processes
PROOF: Suppose that Tn ↑ ∞ are such that G is (H2 , X Tn − )-integrable for all n. If
X Tn − = M + A is the canonical decomposition then
E[(H m )2 · [M ]∞ ]1/2 + E[|H m | · |A|2∞ ]1/2 ≤ E[G 2 · [M ]∞ ]1/2 + E[|G| · |A|2∞ ]1/2 < ∞
Therefore H m are all (H2 , X Tn − )-integrable and hence so is H by the dominated
convergence theorem.
For t 0 fixed
k sup |(H m − H) · X Tn − |k2L 2 ≤ 8k(H m − H) · X Tn − k2H2 → 0
t≤t 0
by the dominated convergence theorem. Whence H m · X Tn − → H · X Tn − uniformly
on [0, t 0 ] in probability. Given " > 0 let n and m be such that P[Tn < t 0 ] < " and
P[((H m − H) · X Tn − )∗t 0 > δ] < ". Whence P[((H m − H) · X )∗t 0 > δ] < 2".
ƒ
4.1.24 Theorem. Let X be a semimartingale. There exists Q ∼ P such that X t ∈
dQ
H2 (Q) for all t and d P is bounded.
4.1.25 Example (Emery). Suppose that T ∼ Exponential(1), P[U = 1] = P[U =
−1] = 12 , and U and T are independent. Let X := U1 t≥T and let the filtration be
the natural filtration of X , so that X is an H2 martingale of finite variation and
[X ] t = 1 t≥T . Let H t := 1t 1 t>0 , which is predictable because it is left continuous.
But H is not (H2 , X )-integrable.
1
2
=∞
E[H · [X ]∞ ] = E
T2
Is H ∈ L(X )? We do know that H · X exists as a pathwise Lebesgue-Stieltjes
integral, namely (H · X ) t = UT 1 t≥T . We have E[|H · X | t ] = ∞ for all t (because the
problem is around zero and not at ∞). It follows in particular that H · X is not a
martingale.
Claim. If S is a finite stopping time with P[S > 0] > 0 then E[|(H · X )S |] = ∞.
Indeed, we have |H · X )S | = T1 1S≥T . The hard part is to show that there is " > 0
such that “{T ≤ "} implies {S ≥ T }”. If this is the case then E[|(H · X )S |] =
E[ T1 1 T ≤" ] = ∞.
(Perhaps use {T ≤ "} = {T ≤ ", S < T } ∪ {T ≤ ", S ≥ T } and 1S<T ∈ F T − =
σ(T ) ∨ N .)
71
Index
(H2 , X )-integrable, 68
H2 -norm, 65
S, 24
indistinguishable, 5
infinitely divisible distribution, 18
integrable processes, 68
integrable variation, 54
intensity, 10
intrinsic Lévy process, 12
adapted, 5
additive process, 12
announcing sequence, 54
arbitrage, 44
arrival rate, 10
Jacod’s countable expansion theorem,
25
Lévy’s theorem, 42
Lévy-Khintchine exponent, 18
Lévy measure, 16
Lévy process, 12
little investment, 44
local martingale, 20
localized class, 20
localizing sequence, 20
Brownian motion, 11
càdlàg, 5
càglàd, 5
canonical decomposition, 58
class D, 52
classical semimartingale, 43
closable, 8
compensator, 57
continuous part of quadratic variation, 32
counting process, 9
modification, 5
monotone vector space, 6
NA, 51
natural process, 54
NFLVR, 44
NFLVR+LI, 44
no arbitrage, 51
no free lunch with vanishing risk, 44
no unbounded profit with bounded
risk, 51
NUPBR, 51
début time, 6
decomposable process, 26
events before time T , 5
events strictly before time T , 5
explosion time, 9
Feller process, 13
Fisk-Stratonovich integral, 39
FLVR, 44
FLVR+LI, 44
free lunch with vanishing risk, 44
optional σ-algebra, 6
optional process, 6
orthogonal, 59
hitting time, 6
Poisson process, 9
72
Index
polarization identity, 31
predictable, 54
predictable variation, 57
prelocally, 68
price process, 44
process of jumps, 5
progressive σ-algebra, 7
progressive process, 7
progressively measurable, 7
purely discontinuous, 59
quadratic pure jump semimartingale,
32
quadratic variation, 31
quasimartingale, 56
random time, 4
semimartingale, 24
semimartingale characteristics, 60
simple predictable process, 24
spatially homogeneous, 13
special semimartingale, 58
square integrable martingale, 9
stable process, 12
stable under stopping, 20
standard Brownian motion, 11
stochastic integral, 24, 28, 66
stochastic integral process, 28
stochastic process, 5
stopped process, 9
stopping time, 4
strategy, 44
subordinator, 12
total semimartingale, 24
total variation process, 21
totally inaccessible, 54
u.c.p., 27
u.i., 7
uniformly integrable, 7
uniformly on compacts in probability, 27
usual conditions, 4
variation, 56
73