Stochastic Analysis I
S.Kotani
April 2006
To describe time evolution of randomly developing phenomena such as motion of particles in random media, variation of stock prices and so on, we have
to treat stochastic processes. The mathematical tools of analyzing stochastic
processes have been well developed from various points of view. The purpose
of this lecture is to give a basic knowledge on Ito’s calculus based on the
martingale theory.
1
1.1
Conditional expectations
σ−fields and information
In the modern probability theory the notion of σ−fields is important because
it is supposed to express a certain aspect of information. Let (Ω, F, P ) be a
probability space, that is, Ω is a set (finite or infinite), F is a σ−field and P
is a probability measure. Recall that a σ−field F is a family of subsets of Ω
satisfying the three properties:
(1) φ, Ω ∈ F
(2) A ∈ F =⇒ Ac ∈ F
∞
S
(3) An ∈ F (n = 1, 2, · · · ) =⇒
An ∈ F.
n=1
Suppose we have two sub σ−fields G1 , G2 of F such that G1 ⊂ G2 . Then this
can be interpreted as
G2 contains more information than G1 .
This is plausible in the following reason. If G1 and G2 consist of finite families
of subsets of Ω, then there exist partitions {A1 , A2 , · · · , Am },{B1 , B2 , · · · , Bn }
of Ω such that

S


G
=
A
;
I
is
arbitrary
subset
of
{1,
2,
·
·
·
,
m}

i
 1
(i∈I
) .
S

 G2 =
Bj ; J is arbitrary subset of {1, 2, · · · , n}


j∈J
Then G1 ⊂ G2 if and only if the partition {B1 , B2 , · · · , Bn } is finer than the
partition {A1 , A2 , · · · , Am } , that is,
any Bj is included in some Ai .
Example 1 Let Ω be the set of all Japanese and consider two partitions (classifications)
of Ω. One classification is by sex and another one is by sex, weight. Define appropriately G1 , G2 associated with these two classifications.
The notion of random variables also should be understood in this context.
A random variable X on (Ω, F, P ) is an F− measurable function on Ω, that
is,
X −1 ((−∞, a]) ≡ {ω ∈ Ω; X(ω) ≤ a} ∈ F for any a ∈ R.
1
Suppose a sub σ-field G of F is generated by a finite partition {B1 , B2 , · · · , Bn }
of Ω. Then a random variable X is G−measurable if and only if
X is a constant on each Bj .
In another word, a finite partition is equivalent to a random variabale taking
only finitely many values. Generally a σ-field σ(X) generated by a random
variable X is defined by
σ(X) = X −1 (F ); F ∈ B (R) ,
where B (R) is the Borel σ-field of R. Analogously a σ-field σ(X1 , X2 , · · · , Xm )
generated by random variables {X1 , X2 , · · · , Xm } is defined by
σ(X1 , X2 , · · · , Xm ) = X−1 (F ); F ∈ B (Rm ) with X = (X1 , X2 , · · · , Xm ).
This σ-field is supposed to contain the information of X.
1.2
Conditional expectations as random variables
In elementary probability theory, for A, B ∈ F, the probability of A conditioned
on B is defined by
P (A ∩ B)
.
P (A |B ) =
P (B)
However in modern probability theory it is supposed that information is nothing
but a σ-field, therefore it is reasonable to define a conditional probability
based on a sub σ-field G of F. For simplicity assume G consists of a finite
partition {B1 , B2 , · · · , Bn } of Ω. Then we define P (A |G ) as a random variable
by
P (A ∩ Bj )
if ω ∈ Bj
P (Bj )
n
X
P (A ∩ Bj )
IBj (ω) ,
=
P (Bj )
j=1
P (A |G ) (ω) =
where IB is the characteristic function of B, that is,
1 if ω ∈ B,
IB (ω) =
.
0 if ω ∈ B c
It should be remarked that P (· |G ) (ω) is a probability measure on (Ω, F) for
each fixed ω ∈ Ω. Analogously the conditional expectation E (X |G ) for a
random variable X with finite expectation is defined by
E IBj X
if ω ∈ Bj
E (X |G ) (ω) =
P (Bj )
n
X
E IBj X
IBj (ω)
=
P (Bj )
j=1
R
Z
n
X
X(ω 0 )P (dω 0 )
Bj
=
IBj (ω) =
X(ω 0 )P (dω 0 |G ) (ω) .
P
(B
)
j
Ω
j=1
2
Therefore the conditional expectation E (X |G ) also is a random variable measurable with respect to G. If X = IA , then E (IA |G ) = P (A |G ) .
In this way, when a sub σ-field G is generated by a finite partition, its
conditional expectation is easily defined. However, if it is not so, we have to
take another approach. To see this, we pay attention to the following properties
of the conditional expectation introduced above. Set Y = E (X |G ) . Then
(1) Y is measurable with respect to G.
(2) Y has a finite expectation and E (IB Y ) = E (IB X) holds for any B ∈ G.
Theorem 2 For any random variable X with finite expectation there exists a
unique random variable Y satisfying (1), (2).
Proof. To prove the uniqueness let Z be another random variable satisfying
(1),(2). Then
E (IB (Y − Z)) = E (IB X) − E (IB X) = 0
for any B ∈ G. Choose B+ = {ω ∈ Ω; Y (ω) − Z(ω) ≥ 0} . Then B+ ∈ G and
IB+ (Y − Z) ≥ 0, hence we have
E IB+ (Y − Z) = 0 =⇒ IB+ (Y − Z) = 0 a.s..
Similarly introducing B− = {ω ∈ Ω; Y (ω) − Z(ω) < 0} , we see IB− (Y − Z) =
0 a.s. Combining these two equalities, we can conclude
Y = Z a.s..
To prove the existence we define a (singed) measure
µ(B) = E (IB X) for B ∈ G.
Then it clearly satisfies
µ(B) = 0 if P (B) = 0,
that is, µ is absolutely continuous with respect to P. Applying the RadonNikodym theorem, we see that there exists a G−measurable random variable Y
with finite expectation such that
Z
µ(B) =
Y (ω)P (dω) = E (IB Y ) ,
B
which comletes the proof.
This unique random variable Y is denoted by E (X |G ) . In addition to the
properties (1),(2), this satisfies
(3)
For any bounded G−measurable random variable G it holds
E (GX |G ) = GE (X |G ) .
(4) X1 ≤ X2 =⇒ E (X1 |G ) ≤ E (X2 |G ), in particular |E (X |G )| ≤ E (|X| |G ) .
(5) G1 ⊂ G2 =⇒ E (E (X |G2 ) |G1 ) = E (X |G1 ) .
(6)
If X is independent to G, then E (X |G ) = EX
3
2
Martingales(discrete time parameter)
2.1
Definitions
The notion of martigales was introduced by Doob in 1950’s, which plays an
indispensable role in the theory of stochastic analysis nowadays. This is a
generalization of sums of indipendent random variables and is effective because
it is flexible under many non-linear transformations.
A family of sub σ−fields {Fn ; n = 0, 1, 2, · · · } of F is called a filtration if
F0 ⊂ F1 ⊂ · · · ⊂ Fn ⊂ · · · ⊂ F.
A stochastic process (one parameter family of random variables)
{Xn }n≥0 = {Xn ; n = 0, 1, 2, · · · }
is called adapted to {Fn }n≥0 , if Xn is measurable with respect to Fn for each
fixed n ≥ 0. An adapted random variables {Xn }n≥0 is called a martingale
with respect to {Fn }n≥0 , if it satisfies
E (|Xn |) < ∞ for n = 0, 1, 2, · · · .
E (Xn+1 |Fn ) = Xn for n = 0, 1, 2, · · · .
(1)
(2)
From this definition martingales can be understood as a mathematical expression of a fair game, which will be ensured in the optional stopping theorem
below.
Example 3 Let {Yn }n≥1 be a family of independent random variables with finite expectations.
Xn = (Y1 − EY1 ) + (Y2 − EY2 ) + · · · + (Yn − EYn ) for n ≥ 1
(i)
X0 = 0
is a martingale with respect
to Fn = σ(X1 , X2 , · · · , Xn ) and F0 = {φ, Ω} .
2
(ii) Moreover assume E |Yn | < ∞ and EYn = 0 for n ≥ 1. Then
2
Xn = (Y1 + Y2 + · · · + Yn ) − EY12 + EY22 + · · · + EYn2
X0 = 0
for n ≥ 1
is a martingale.
(iii) Suppose E (Yn ) = 1 for all n ≥ 0. Then
Xn = Y1 Y2 · · · Yn−1 Yn for n ≥ 1
X0 = 1
is a martingale.
Example 4 Let X be a random variable with finite expectation and {Fn }n≥0
be a filtration. Then
Xn = E (X |Fn )
becomes a martingale.
4
Example 5 (martingale transformation=discrete stochastic integral)
Let {Mn }n≥0 be a martingale and {fn }n≥0 be random variables such that fn is
bounded and measurable with respect to Fn for n = 0, 1, 2, · · · . Then
 n−1
 P
fk (Mk+1 − Mk ) if n ≥ 1
Xn =
 k=0
0
if n = 0.
becomes a martingale.
It is convenient to introduce wider notions. {Xn }n≥0 is called a submartingale(supermartingale) with respect to {Fn }n≥0 , if in addition to the property (1) it satisfies
(3)
E (Xn+1 |Fn ) ≥ Xn for n ≥ 0.
(E (Xn+1 |Fn ) ≤ Xn for n ≥ 0.)
Example 6 Let {Xn }n≥0 be a martingale and f be a convex function, that is,
f (αx + βy) ≤ αf (x) + βf (y) if α + β = 1, α, β ≥ 0 and x, y ∈ R .
Suppose E (|f (Xn )|) < ∞ for n = 0, 1, 2, · · · . Then {f (Xn )}n≥0 becomes a
submartingale.
Example 7 Let {Yn }n≥0 be a Markov process on S with transition kernel {p(x, dy)}
and f be a subharmonic function on S for this Markov process, that is,
Z
f (x) ≤
f (y)p(x, dy).
S
If E (|f (Yn )|) < ∞, then Xn = f (Yn ) is a submartingale.
2.2
Martingale inequality
In order to establish the theorem of law of large numbers in its full generality,
Kolmogorov used a maximal inequality for sums of independent random variables with 0 means as an analogy of an inequality in Fourier analysis. Doob
pointed out the inequality is valid also for submartingales, and it became a very
useful tool in stochastic analysis.
Theorem 8 (martingale inequaly) Let {Xn }n≥0 be a submartingale. Then
for any λ > 0 it holds that
−1
P max Xk ≥ λ ≤ λ E Xn ; max Xk ≥ λ ≤ λ−1 E Xn+ .
0≤k≤n
0≤k≤n
Proof. Since the last two inequalities are trivial, we show only the first one.
Set


A = max Xk ≥ λ
.
0≤k≤n

Am = {X0 < λ, X1 < λ, · · · , Xm−1 < λ, Xm ≥ λ}
5
Then {Am }0≤m≤n are disjoint and
A=
n
[
Am , Am ∈ Fm .
m=0
The definition of submartingales((3) of Section2.1) implies for n ≥ m
E(Xn ; Am ) ≥ E(Xm ; Am ).
Hence
E(Xn ; A) =
n
X
n
X
E(Xn ; Am ) ≥
m=0
E(Xm ; Am ) ≥ λP (A).
m=0
Corollary 9 Suppose {Xn }n≥0 is a martingale. Then
p
(i)
P max |Xk | ≥ λ ≤ λ−p E (|Xn | ) for p ≥ 1 (Kolmogorov-Doob),
0≤k≤n
p
p
p
p
(ii) E max |Xk | ≤
E (|Xn | ) for p > 1.
0≤k≤n
p−1
p
p
Proof. Since f (x) = |x| is convex if p ≥ 1, {|Xn | }n≥0 becomes a submartingale due to Example8. Applying Theorem11 , we see
p
p
P max |Xk | ≥ λ = P max |Xk | ≥ λp ≤ λ−p E (|Xn | ) .
0≤k≤n
0≤k≤n
To prove (ii), set
Y = max |Xk | .
0≤k≤n
Observe for a non-negative random variable Y
Z ∞
Z
p
p−1
EY = E
pλ IY ≥λ dλ =
0
∞
pλp−1 P (Y ≥ λ) dλ
0
holds. Thus from (i) it follows that
Z ∞
p
EY ≤
pλp−1 λ−1 E (|Xn | ; Y ≥ λ) dλ
0
p
E |Xn | Y p−1
p−1
−1
p
p p
1−p−1
≤
{E (|Xn | )}
{EY p }
,
p−1
=
hence
EY p ≤
p
p−1
6
p
p
E (|Xn | ) .
2.3
Optional stopping theorem
For further investigation of stochastic processes, the notion of stopping times
is crucial. A stopping time is not simply a non-negative random variable, but
it has to be non-anticipating, which will be specified soon. Let {Fn }n≥0 be a
filtration. A non-negative integer valued random variable τ taking possibly ∞
is called a stopping time with respect to {Fn }n≥0 if it satisfies
{τ ≤ n} ∈ Fn for n ≥ 0 (⇐⇒ {τ = n} ∈ Fn for n ≥ 0) .
Typical examples of stopping times are given by hitting times for adapted
stochastic processes.
Example 10 Let {Xn }n≥0 be a stochastic process adapted to {Fn }n≥0 and F ∈
B(R). Define
inf {n ≥ 0; Xn ∈ F } if Xn ∈ F for some n < ∞
τ=
.
∞
otherwise
Then τ is called the first hitting time to F and becomes a stopping time with
respect to {Fn }n≥0 . On the other hand, the last hitting time
τ=
sup {n ≥ 0; Xn ∈ F } if Xn ∈ F for some n < ∞
,
∞
otherwise
can not be a stopping time.
We remark the following
Lemma 11 Let σ, τ be stopping times. Then σ ∧ τ, σ ∨ τ and σ + τ are also
stopping times.
Proof. We prove the statement only for σ + τ. Observe
{σ + τ = n} =
n
[
{σ = k, τ = n − k} ∈ Fn ,
k=0
which concludes the proof.
Now the theorem is
Theorem 12 (Optional stopping theorem) Let {Xn }n≥0 be a submartingale. Then for any stopping times τ, σ such that τ ≥ σ, {Xn∧τ − Xn∧σ }n≥0
becomes a submartingale. In particular, if τ is bounded, then
E (Xτ ) ≥ E (Xσ ) .
Proof. Set fk = Iτ >k≥σ = Iτ >k − Iσ>k . Then
Xn∧τ − Xn∧σ =
n−1
X
fk (Xk+1 − Xk ) if n ≥ 1,
k=0
7
and, since fk ≥ 0, we see as in Example5 that {Xn∧τ − Xn∧σ }n≥0 is a submartingale:
! n−1
n
X
X
E
fk (Xk+1 − Xk ) |Fn =
fk (Xk+1 − Xk ) + fn E (Xn+1 − Xn |Fn )
k=0
k=0
≥
n−1
X
fk (Xk+1 − Xk ) ,
k=0
which completes the proof.
The theorem says that if {Xn } is a martingale and τ is a bounded stopping
time, then
E (Xτ ) = E (X0 ) .
In this sense, martingales are fair games.
3
Martingale convergence theorems
One advantage of using martingales in analysis of stochastic processes is that
they make the proofs of convergences easy. The following argument initiated by
Doob is a key to establish the convergence theorems for (sub)martingales.
For a sequence of real numbers {x0 , x1 , · · · , xN } and (a < b), set τ0 = 0 and
τn+1 = min {k ≥ τn ; xk ≤ a}
for even n
τn+1 = N
if {k ≥ τn ; xk ≤ a} = φ
τn+1 = min {k ≥ τn ; xk ≥ b}
for odd n .
τn+1 = N
if {k ≥ τn ; xk ≥ b} = φ
Notice here
1 = τ0 ≤ τ1 ≤ τ2 ≤ · · · ≤ τN = N and, if τn = N for some n < N,
then τn+1 = τn+2 = · · · = τN = N.
Define the upcrossing number of {x0 , x1 , · · · , xN } between (a, b) by
max k; xτ2k−1 ≤ a, xτ2k ≥ b U = U (x0 , x1 , · · · , xN ; (a, b)) =
.
0 if k; xτ2k−1 ≤ a, xτ2k ≥ b = φ
We illustrate those definitions below.
Lemma 13 (Doob) Let {X0 , X1 , X2 , · · · , XN } be a submartingale and U be its
upcrossing number between (a, b). Then
+
(b − a) E (U ) ≤ E (XN ∨ a − X0 ∨ a) ≤ E (XN − X0 ) .
8
Proof. For simplicity assume N is even. Observe
XN − X0 = (Xτ1 − Xτ0 ) + (Xτ2 − Xτ1 ) + · · · + XτN − XτN −1
≥ (b − a) U + (Xτ1 − Xτ0 ) + (Xτ3 − Xτ2 ) + · · · + XτN +1 − XτN .
Since {τn } are stopping times, applying Theorem12, we see
E Xτ2n+1 − Xτ2n ≥ 0.
Thus we have
E (XN − X0 ) ≥ (b − a) E (U ) .
Set Yn = Xn ∨ a. Then it is easy to see that {Yn } is also a submartingale and
U (X0 , X1 , · · · , XN ; (a, b)) = U (Y0 , Y1 , · · · , YN ; (a, b)),
we complete the proof.
Now the main result is
Theorem 14 Suppose {Xn }n≥0 is a submartingale satisfying
E Xn+ ≤ C for all n ≥ 0 (x+ = x ∨ 0)
with some C > 0. Then there exists a finite random variable X∞ such that
lim Xn = X∞ a.s., and E (|X∞ |) < ∞
n→∞
hold.
Proof. For a < b, UN (ω) = U (X0 (ω) , X1 (ω) , · · · , XN (ω) ; (a, b)) % ∞ as
N → ∞. Define
U (ω) = lim UN (ω) .
N →∞
Since
+
+
E (XN − X0 ) ≤ E XN
+ E (|X0 |) ≤ C + E (|X0 |) ,
from Lemma19 it follows that
EU ≤ C + E (|X0 |) , hence U (ω) < ∞ a.s..
However setting
Aa,b
= ω ∈ Ω; lim inf Xn (ω) < a < b < lim sup Xn (ω) ,
n→∞
n→∞
we see U (ω) = ∞ for ω ∈ Aa,b . Hence we have P (Aa,b ) = 0 from (1). Set
[
A=
Aa,b .
a<b
a,b∈Q
Then P (A) = 0 and for ω ∈ Ac , we have
lim inf Xn (ω) = lim sup Xn (ω) ≡ X∞ (ω) ∈ [−∞, +∞].
n→∞
n→∞
9
(1)
However Fatou’s lemma implies
E (|X∞ |) ≤ lim inf E (|Xn |) ≤ 2C + E (|X0 |) < ∞.
n→∞
Here we have used
E (|Xn |) = 2E Xn+ − E (Xn ) ≤ 2E Xn+ − E (X0 ) ,
which completes the proof.
As for supermartingales we have
Theorem 15 Let {Fn }−∞<n≤0 be a family of σ−fields such that
· · · ⊂ Fn ⊂ Fn+1 ⊂ · · · ⊂ F−1 ⊂ F0 ,
and {Xn }−∞<n≤0 be a family of random variables adapted to {Fn } . Then there
exists a random variable X−∞ ∈ (−∞, +∞] such that
lim Xn = X−∞
n→−∞
a.s..
Moreover if
lim EXn < ∞,
n→−∞
then
E (|X−∞ |) < ∞, hence |X−∞ | < ∞.
Proof. The number of downcrossings D of sequences {XM , XM +1 , · · · , X−1 , X0 }
between (a, b) can be defined analogously as the number of upcrossings and we
can prove the following estimate
(b − a) E (D) ≤ E (XM ∧ b − X0 ∧ b) ≤ |b| + E (|X0 |) ,
because {−Xn } is a submartinagle and its upcrossing number between (−b, −a)
is eaqual to the downcrossing number for {XM , XM +1 , · · · , X−1 , X0 } . Now the
rest of the proof is the same as that of Theorem14 except that X−∞ may take
+∞. However if lim EXn < ∞ is valid, then
n→−∞
E (|Xn |) = EXn + 2EXn− ≤ EXn + 2EX0−
=⇒ E (|X−∞ |) ≤ lim EXn + 2EX0− < ∞.
n→−∞
Corollary 16 Let {Fn }−∞<n<+∞ be a family of σ−fields such that
· · · ⊂ Fn ⊂ Fn+1 ⊂ · · · ,
and X be a random variable with finite expectation. Then
E (X |F+∞ ) as n → +∞
E (X |Fn ) →
E (X |F−∞ ) as n → −∞
almost surely and in L1 (Ω, P ), where
!
[
F+∞ = σ
Fn ,
!
F−∞ = σ
n
\
n
10
Fn
.
4
Martingales(continuous time parameter)
In this section we discuss martingales with continuous time parameter. Let
{Ft }t≥0 be a filtration with continuous parameter, that is,
Fs ⊂ Ft if 0 ≤ s ≤ t.
Let {Xt }t≥0 be a stochastic processes (a family of random variables) adapted
to {Ft } , that is, for each t ≥ 0 Xt is Ft −measurable. We can define the
notions of martingales, submartingales and supermartingales in this case just
like the discerete case. In the following arguments we need to impose the right
continuity of {Xt }t≥0 , that is,
P lim Xs = Xt for every t ≥ 0 = 1
s↓t
Throughout this lecture we assume that all stochastic processes are at least
right continuous. For a filtration {Ft } set
\
\
Ft+0 =
Fs =
Ft+ n1 ⊃ Ft .
s>t
n≥1
Then we have
Lemma 17 Let {Xt }t≥0 be a right continuous submartingale with respect to
{Ft } . Then {Xt }t≥0 is a submartingale with respect to {Ft+0 } .
Proof. Let t > s ≥ 0 and choose n ≥ 1 such that t > s + n1 . Then
E Xt Fs+ n1 ≥ Xs+ n1 .
The right hand side converges to Xs as n → ∞. On the other hand, the left
hand side converges to E (Xt |Fs+0 ) due to Corollary16, which completes the
proof.
Keeping this Lemma in our minds, from now on we assume filtrations are
always right continuous, that is,
Ft+0 = Ft for every t ≥ 0.
(2)
The three main results for martingales with discrete parameter are also valid
for martingales with continuous parameter.
Theorem 18 (Martingale inequalities)
(i) Let {Xt }t≥0 be a right continuous submartingale. Then for any λ > 0
P sup Xs ≥ λ ≤ λ−1 E Xt ; sup Xs ≥ λ ≤ λ−1 E Xt+ ≤ λ−1 E (|Xt |) .
0≤s≤t
0≤s≤t
Suppose {Xt }t≥0 is a right continuous martingale. Then
p
(ii) P sup |Xs | ≥ λ ≤ λ−p E (|Xt | ) for p ≥ 1, λ > 0 ,
0≤s≤t
p
p
p
p
(iii) E sup |Xs | ≤
E (|Xt | ) for p > 1.
p
−
1
0≤s≤t
11
Proof. Let Q be the set of all rational numbers and Fn be finite subsets of
Q ∩ [0, T ] such that
Fn ⊂ Fn+1 , max Fn = t for every n ≥ 1 and
∞
[
Fn = Q ∩ [0, T ].
n=1
Since {Xt }t∈Fn is a submartingale with discrete parameter, we can apply Theorem8 and Corollary9. The right continuity of {Xt } implies
max Xs ↑ sup Xs as n → ∞,
s∈Fn
s∈[0,T ]
which shows (i). (ii) and (iii) can be proved similarly.
To prove the optional stopping theorem we give a remark on stopping times.
The notion of stopping times can be defined similarly, that is, a non-negative
random variable τ ≤ ∞ is called an {Ft } −stopping time if
{τ ≤ t} ∈ Ft for every t ≥ 0.
The right continuity of {Ft } implies this definition is equivalent to
{τ < t} ∈ Ft for every t > 0,
because

∞ S

τ ≤ t − n1 ∈ Ft
 {τ < t} =
n=1
.
∞ T

 {τ ≤ t} =
τ < t + n1 ∈ Ft+0
n=1
We give examples.
Example 19 Let {Xt }t≥0 be a right continuous stochastic process adapted to
{Ft } . For A ⊂ R set
inf {t ≥ 0; Xt ∈ A} if {·} 6= φ
τA =
∞
if {·} = φ.
If A is an open set or a closed set, then τA becomes an {Ft } −stopping time.
Proof. Assume first A is open. Then
∞
[
{τA < t} =
{Xr ∈ A} ∈ Ft .
r∈Q, 0≤r<t
Suppose A is closed. Then
{τA ≤ t} =
∞
\
{τAn ≤ t} ∈ Ft ,
n=1
where
An =
1
x ∈ R ; dist (x, A) >
(open set in R) .
n
For a general Borel set A, the measurability of τA is a highly non-trivial problem, and we need extra properties for {Xt }t≥0 and the completion of probability
spaces.
12
Lemma 20 (i) Let σ, τ be stopping times. Then so are
σ ∧ τ, σ ∨ τ, σ + τ.
(ii) Let {σn }n≥1 be a sequence of stopping times which are decreasing or increasing. Then so is lim σn .
n→∞
Proof. We left the proof for the readers.
Theorem 21 (Optional stopping theorem) Let {Xt }t≥0 be a right continuous
submartingale and σ, τ be stopping times such that τ ≥ σ. Assume
E sup |Xt | < ∞ for each T > 0.
(3)
0≤t≤T
Then {Xt∧τ − Xt∧σ } becomes a submartingale. In particular, if τ is bounded,
we have
E (Xτ ) ≥ E (Xσ ) .
Proof. For fixed n ≥ 1, set


 τe = k + 1
2n
k+1

 σ
e=
2n
k
k+1
≤τ <
2n
2n
k
k+1 ,
if n ≤ σ <
2
2n
if
and
Fek = F kn .
2
n o
Then 2n τe, 2n σ
e are Fek − stopping times, because
k
{2 τe ≤ k} = τ < n ∈ F kn = Fek .
2
2
n
n o
ek = X k is a submartingale with respect to Fek , we see from
Since X
n
2
Theorem12 for k > l
ek∧2n τe − X
ek∧2n σe Fel ≥ X
el∧2n τe − X
el∧2n σe .
E X
k
l−1
l
e
Now assume k−1
2n ≤ t < 2n , 2n ≤ s < 2n and let A ∈ Fs ⊂ Fl . Then the above
inequality implies
E X kn ∧eτ − X kn ∧eσ IA ≥ E X ln ∧eτ − X ln ∧eσ IA ,
2
2
2
On the other hand, since τe ↓ τ, σ
e ↓ σ and
continuity of {Xt } shows
k
2n
↓t,
2
l
2n
↓ s as n → ∞, the right
→ Xt∧τ , X ln ∧eτ → Xs∧τ , X kn ∧eσ → Xt∧σ , X ln ∧eσ → Xs∧σ as n → ∞.
2
2
2
However, since X kn ∧eτ , X kn ∧eσ ≤ sup0≤u≤t+1 |Xu | , which is integrable due
2
2
to the assumption (3) , the dominated convergence theorem proves
X
k
2n
∧e
τ
E ((Xt∧τ − Xt∧σ ) IA ) ≥ E ( (Xs∧τ − Xs∧σ ) IA ) for A ∈ Fs .
13
n
o
Remark 22 Since a more detailed argument shows that X kn ∧eτ , X kn ∧eσ are
2
2
uniformly integrable, we see that the condition (3) is unnecessary in general.
Theorem 23 (Martingale convergence theorem) If {Xt }t≥0 is a right continuous submartingale satisfying
E Xt+ ≤ C for any t ≥ 0,
then there exists a finite random variable X∞ such that
Xt → X∞ as t → ∞ a.s. and E (|X∞ |) < ∞.
Proof. For fixed n ≥ 1, set
Tn =
k
; k = 0, 1, 2, · · ·
2n
, T =
∞
[
Tn .
n=1
n
Then T is countable and dense in [0, ∞). For N ≥ 1 and a < b, let UN
(a, b) be
the upcrossing number of {Xt }t∈Tn ∩[0,N ] between (a, b). Then from Lemma13,
we have
+
n
(b − a) E (UN
(a, b)) ≤ E XN
+ E (|X0 |) ≤ C + E (|X0 |) .
n
Since as n, N → ∞, UN
(a, b) is increasing, let U (a, b) be its limit. Then we see
E (U (a, b)) ≤
C + E (|X0 |)
< ∞,
b−a
which implies
U (a, b) < ∞ a.s..
Varying a < b among all rational numbers, we see that with probability 1
lim Xt ∈ [−∞, +∞]
t→∞
t∈T
exists a.s. .
The right continuity of {Xt } implies
lim Xt = lim Xt .
t→∞
t∈T
t→∞
On the other hand, we have
E (|Xt |) = 2E Xt+ − EXt ≤ −EX0 + 2E Xt+ ≤ −EX0 + 2C,
hence Fatou’s lemma shows
E (|X∞ |) ≤ −EX0 + 2C < ∞.
14
5
Brownian motion
A typical and the most important martingale with continuous parameter is
Brownian motion. In this section we introduce this process and study its basic
properties. Brownian motion was found by an English botanist Brown in 1828
as a random motion of some components of pollen. In 1905 Einstein used this
process to study the existence of molecules without knowing the previous discovery by Brown. Brownian motion was first defined mathematically as a continuous stochastic process by Wiener in 1923, and this process is called sometimes
Wiener process. Lévy studied properties of paths of Brownian motions from
very original points of view. Kolmogorov tried to develop a dynamical theory
of Markov processes and pointed out that Brownian motion could play a basic
role for its purpose. Ito initiated a random dynamical theory based on Brownian motion in 1942 and started a rigorous treatment of stochastic differential
equations.
A stochastic process {Bt }t≥0 on a probability space (Ω, F, P ) is called a
Brownian motion if it satisfies
(1) Each sample {Bt }t≥0 is continuous as a function of t ≥ 0.
(2) For any sequence {0 ≤ t0 ≤ t1 ≤ · · · ≤ tn } , the random variables
Bt1 − Bt0 , Bt2 − Bt1 , · · · , Btn − Btn−1
are independent.
(3) The distribution of Bt − Bs is N (0, |t − s|) , that is, a Gaussian distribution
with expectation 0 and variance |t − s| .
These three properties determine uniquely the distribution of a Brownian motion
as a stochastic process. These three properties are not independent. Actually
owing to the central limit theorem the properties (1) and (2) imply that the
distribution of Bt − Bs becomes a Gaussian.
The existence of a stochastic process satisfying the three properties is not
trivial, and we had to wait Wiener. He constructed a Brownian motion as a
Fourier series with random coefficients. Now a more intuitive introduction of
the process is possible as a limit of a simple random walk. We give here its
short explanation. Let {Yn }n≥1 be independent random varibales taking values
{±1} with equal probability 1/2. Set
Y1 + X2 + · · · + Xn if n ≥ 1,
Xn =
0
if n = 0.
Define a continuous stochastic process {Xt }t≥0 by
Xt = (t − n) Xn+1 + (n + 1 − t) Xn
2
2
if n ≤ t ≤ n + 1.
Since EXt = 0 and E (Xt ) = (t − n) + n if n ≤ t ≤ n + 1, we see
2
2
(mt − n) + n
Xmt
√
=
if n ≤ mt ≤ n + 1.
E
m
m
n
Denoting t =
+ ε, we have
m
2
Xmt
n
E √
= mε2 + .
m
m
15
However, 0 ≤ ε ≤
1
, therefore as m → ∞ it holds that
m
2
Xmt
→t.
E √
m
√
Then the central limit theorem tells us that the distribution of {Xmt / m}
converges to N (0, t) as m → ∞. If we take a closer look at vector valued random
variables
Xmtn − Xmtn−1
Xmt1 − Xmt0 Xmt2 − Xmt1
√
√
√
,
,··· ,
,
m
m
m
we can show that their distributions converge to independent Gaussian distibutions N (0, t1 − t0 ), N (0, t2 −
√t1 ), · · · , N (0, tn − tn−1 ). To construct a Brownian
motion as a limit of {Xmt / m} this argument is not sufficient
and we have to
√
discuss the convergence of the distributions µm of {Xmt / m} induced on the
space
C ([0, ∞)) = {w; w is a continuous function on [0, ∞)} .
Donsker established this convergence and the limit µ is a probability measure
on C ([0, ∞)) governing the probability law of the Brownian motion. Although
its sample path is continuous, it is known that any sample path is differentiable
at no points of [0, ∞), which makes it difficult to construct a stochastic analysis
based on Brownian motions. To readers who get interested in a more detailed
explanation of the construction of Brownian motions, we recomend them to look
suitable text books.
A Brownian motion in d-dimensional space
Rd (d−dim.B.M. in short) can be
defined similarly. Let Bt = Bt1 , Bt2 , · · · , Btd t≥0 be an Rd −valued stochastic
process satisfying
(1) Each sample {Bt }t≥0 is continuous as a function of t ≥ 0.
(2) For any sequence {0 ≤ t0 ≤ t1 ≤ · · · ≤ tn } , the random variables
Bt1 − Bt0 , Bt2 − Bt1 , · · · , Btn − Btn−1
are independent.
(3) The distribution of Bt −Bs is N (0, |t − s| I) , that is, a Gaussian distribution
with expectation 0 and variance matrix |t − s| I.
From this definition we can easily see that each Bti t≥0 is a 1-dim.B.M. and
n o
Bt1 t≥0 , Bt2 t≥0 , · · · , Btd t≥0 are independent.
Set
Ft = σ − {Bs ; 0 ≤ s ≤ t} .
We remark
Proposition 24 For a d−dim.B.M.{Bt }t≥0 ,
are martingales with respect to {Ft } .
16
Bti
t≥0
and
n
o
Bti Btj − δij t
t≥0
Proof. For t ≥ s ≥ 0,
E Bti |Fs = E Bti − Bsi |Fs + E Bsi |Fs
= E Bti − Bsi + Bsi = Bsi .
We have used here the fact that Bti − Bsi is independent of Fs . On the other
hand,
E(Bti Btj |Fs )
= E Bti − Bsi (Btj − Bsj ) |Fs + E Bsi (Btj − Bsj ) |Fs + E Bti Bsj |Fs
= E Bti − Bsi (Btj − Bsj ) + Bsi E(Btj − Bsj ) + Bsj E Bti |Fs
= δij (t − s) + Bsi Bsj .
A Brownian motion has a lot of symmetries. The following proposition
reveals a part of these symmetries.
Proposition 25 Let {Bt }t≥0 be a d-dim.B.M. starting from 0. Then
√
(i) For a fixed c > 0, {Bct / c}t≥0 is a d-dim.B.M. starting from 0.
(ii) For a fixed t0 ≥ 0, {Bt+t0 − Bt0 }t≥0 is a d-dim.B.M. starting from 0.
(iii) {tBt−1 }t≥0 is a d-dim.B.M. starting from 0.
Proof. Only (iii) is not trivial to prove. To show this, we have only to look
the property
(2) for B̂t =otBt−1 . To see this, we consider a family of random
n
vectors B̂t0 , B̂t1 , · · · , B̂tn . It is easy to see that their joint distribution is a
Gaussian
in R(n+1)d with expectation 0. If we could show that the distributions
of B̂t0 , B̂t1 , · · · , B̂tn , (Bt0 , Bt1 , · · · , Btn ) are equal, then we have the property
n o
(2) for B̂t
. However Gaussian distributions in multi-dimensional space are
t≥0
determined by their expectations and covariances, we have only to compute for
t ≥ s ≥ 0 and 1-dim.B.M.
E B̂t B̂s = tsE (Bt−1 Bs−1 ) = ts × t−1 ∧ s−1 = s = E (Bt Bs ) .
6
Stochastic integral
To define an integral based on a function Y on the real line, usually the function
has to be of bounded variation. However, as we see in Brownian motions,
martingales does not have paths with bounded variation. Therefore the ordinary
Lebesgue-Stieltjes integral can not be applied to an integration with respect to
martingales. Fortunately martingales have an extended version of variation,
which will be seen in this section, and we make use of this property for the
definition of the stochastic integral.
17
6.1
Variational processes
Let {0 = t0 < t1 < t2 < · · · < tn = T } be a partition of [0, T ] and denote it by
Π. For p ≥ 1 and a function {Yt } on [0, T ] set
Vp (Π, Y ) =
n−1
X
Yt
i+1
p
− Yti , V1 (Y ) = sup V1 (Π, Y ).
Π
i=0
A function {Yt }t≥0 is called of bounded variation on [0, T ] if V1 (Y ) < ∞. Now,
denoting kΠk = max {ti+1 − ti ; 0 ≤ i ≤ n − 1} , we have for p > 0
p
Vp+1 (Π, Y ) ≤ V1 (Π, Y ) ×
p
|Yt − Ys | ≤ V1 (Y ) ×
sup
0≤s≤t≤T
t−s≤kΠk
sup
|Yt − Ys | .
0≤s≤t≤T
t−s≤kΠk
Therefore, if {Yt }t≥0 is continuous and of bounded variation, then Vp+1 (Π, Y ) →
0 as kΠk → 0.
What we would like to show in this section is the existence of non-decreasing,
continuous function At for square integrable martingale M such that V2 (Πn , M ) →
AT if kΠn k → 0. Then a decomposition:
Mt2 = Nt + At
(4)
is valid, where {Nt }t≥0 is a continuous martingale. The {At }t≥0 is called the
variational process associated with {Mt }t≥0 . This process is crucial to de
fine stochastic integrals based on martingales. Since Mt2 is a submartingale,usually this process is introduced through the Doob-Meyer decomposition
of submartingales. However in this lecture we employ another approach to take
a shortcut.
Fix a right continuous filtration {Ft }t≥0 throughout this section. A stochastic process {ft }t≥0 adapted to the given filtration is called a stepfunction if there
exists an increasing sequence {tn }n≥0 and random variables {φn }n≥0 such that
0 = t0 < t1 < t2 < · · · < tn < · · · , tn → ∞ as n → ∞,
ft = φi
if t ∈ (ti , ti+1 ], where φi is measurable w.r.t. Fti
In this section we assume the boundedness for stepfunctions, that is, there exists
a constant C such that
|ft (ω)| ≤ C
for all t ≥ 0 and ω ∈ Ω.
For a later purpose we denote the set of all bounded stepfunctions by L0 =
L0 (F· ). Let {Mt }t≥0 be a continuous martingale which is square integrable,
that is, EMt2 < ∞ for every t ≥ 0 and satisfies M0 = 0. We define a stochastic
integral It (f ) of a stepfunction based on this martingale by I0 (f ) = 0 and
It (f ) =
k−1
X
φi ∆i + φk ∆t if tk < t ≤ tk+1 ,
(5)
i=0
where ∆i = Mti+1 − Mti , ∆t = Mt − Mtk . Since there are many ways to express
a stepfunction, it should be proved that the above integral does not depend on
the expressions. We leave its proof to the readers.
18
Lemma 26 {It (f )}t≥0 is a continuous martingale satisfying for t > s ≥ 0
k−1
X
E It (f )2 |Fs = Is (f )2 + E
i=l
!
φ2i ∆2i + φ2k ∆2t Fs ,
(6)
where we assume tl = s without loss of generality.
Proof. The continuity is clear from the definition. A computation
!
k−1
X
E (It (f ) |Fs ) = E (Is (f ) |Fs ) + E
φi ∆i + φk ∆t Fs = Is (f ),
i=l
shows the martingale property of {It (f )} . Now we calculate the conditional
expectation of the square:

!2 
k−1
X
E It (f )2 |Fs = E  Is (f ) +
φi ∆i + φk ∆t Fs 
i=l
!
!
k−1
k−1
X
X
2
2 2
2 2
= Is (f ) + E
φi ∆i + φk ∆t Fs + 2Is (f )E
φi ∆i + φk ∆t Fs
i=l
i=l


X
X

+ 2E
φi φi ∆i ∆j +
φi φk ∆i ∆t Fs 
l≤i<j≤k−1
l≤i≤k−1
2
= Is (f ) + E
k−1
X
!
φ2i ∆2i
+
φ2k ∆2t
,
i=l
because for i < j we see
E ( φi φi ∆i ∆j | Fs ) = E E φi φi ∆i ∆j Ftj Fs
= E φi φi ∆i E ∆j Ft Fs = 0,
j
and
E ( φi φk ∆i ∆t | Fs ) = E ( E (φi φk ∆i (Mt − Mtk ) |Ftk )| Fs )
= E ( φi φk ∆i E (Mt − Mtk |Ftk )| Fs ) = 0,
which proves (6).
Since we have to discuss convergence of sequences of continuous martingales,
let M2T = M2T (F· ) be the set of all continuous square integrable martingales
on [0, T ] and introduce a norm of M = {Mt }0≤t≤T ∈ M2T
kM k = kM k2,T =
q
E (MT2 ).
Lemma 27 With this norm the space M2T becomes a Hilbert space.
19
Proof. For M = {Mt }0≤t≤T ∈ M2T , we see
Mt = E (MT |Ft ) if t ≤ T.
Therefore if we have a Cauchy sequence {M n }n≥1 ∈ M2T , whose {MTn } converges to an f ∈ L2 (Ω, P ) in L2 −norm, then a martingale defined by
Mt = E (f |Ft )
becomes a limit of {M n }n≥1 in M2T if we could show the continuity of {Mt } .
To see this we apply (ii) of Theorem18 to a martingale M n − M m , that is,
2
P
sup |Mtn − Mtm | ≥ λ ≤ λ−2 E |MTn − MTm |
(7)
0≤t≤T
for any λ > 0. Since {MTn }n≥1 is a Cauchy sequence, for any k ≥ 1 we can
choose a subsequence {nk }k≥1 such that
2 1
n
E MT k+1 − MTnk ≤ k .
8
Set
Ak = ω ∈ Ω;
n
1
sup Mt k+1 (ω) − Mtnk (ω) ≥ k
2
0≤t≤T
, A=
\ [
Ak .
n≥1k≥n
Then (7) implies P (Ak ) ≤ 2−k for every k ≥ 1, thus


[
X
X 1
P (A) ≤ P 
Ak  ≤
P (Ak ) ≤
→ 0 as n → ∞,
2k
k≥n
k≥n
k≥n
hence we see P (A) = 0. If ω ∈
/ A, then there exists a K ≥ 1 such that
n
1
sup Mt k+1 (ω) − Mtnk (ω) ≤ k
2
0≤t≤T
for any k ≥ K.
For k > l, we have for every t ∈ [0, T ]
|Mtnk (ω) − Mtnl (ω)| ≤
X
ni+1
Mt
(ω) − Mtni (ω) ≤
l≤i≤k−1
X
l≤i≤k−1
1
→0
2i
{Mtnk
as k, l → ∞, hence
(ω)}0≤t≤T becomes a Cauchy sequence with sup-norm
and converges uniformly on [0, T ] to a continuous function {Mt0 (ω)}0≤t≤T . Since
Mtn = E (MTn |Ft ) , we see
2
2
|Mtn − Mtm | ≤ E |MTn − MTm | |Ft
2
2
=⇒ E |Mtn − Mtm | ≤ E |MTn − MTm | → 0
as n, m → ∞, thus for every t ∈ [0, T ], {Mtn } becomes also a Cauchy sequence
converging to Mt , which concludes Mt = Mt0 .
The Lemma below indicates that continuous martingales can not be functions of bounded variation unless they are constants.
20
Lemma 28 Suppose {Xt }t≥0 is an adapted continuous process and has a decomposition
Xt = Mt + At
with a continuous martingale {Mt } and a bounded variation process {At } with
A0 = 0. Under the condition that E (V1 (A)) < ∞, this decomposition is unipue.
Proof. Suppose we have another such decomposition {Mt0 , A0t } of {Xt }t≥0 and
set
Yt ≡ Mt − Mt0 = A0t − At ,
{Yt }t≥0 is a continuous martingale with bounded variation. Here, first we assume {Yt }t≥0 is bounded by C. Then observing
V2 (Π, Y ) ≤ 2CV1 (Y ), V1 (Y ) ≤ V1 (A) + V1 (A0 ) ∈ L1 ,
we have by the dominated convergence theorem
E (V2 (Π, Y )) → 0 as kΠk → 0.
(8)
On the other hand, from the martingale property for t > s ≥ 0 it follows that
2
E (Yt − Ys ) = E Yt2 − 2Yt Ys + Ys2 = E Yt2 − Ys2 ,
hence we see
E (V2 (Π, Y )) =
n−1
X
E Yti+1 − Yti
2
i=0
=
n−1
X
E Yt2i+1 − Yt2i = EYT2 .
i=0
EYT2
= 0, which is nothing but Yt = 0 for every
This together with (8) shows
t ≥ 0 from the martingale property of {Yt } . If {Yt }t≥0 is not bounded, we have
only to stop {Yt } by
τc = inf {t ≥ 0; |Yt | > c} ,
then {Yt∧τc } becomes a martingale, because it satisfies (3):
E sup |Yt | ≤ E (V1 (Y )) ≤ E (V1 (A) + V1 (A0 )) < ∞.
0≤t≤T
Applying the above argument to {Yt∧τc } , we see Yt∧τc = 0. The rest of the
proof is clear.
Now, for {Mt }t≥0 ∈ M2T and a fixed partition
Π = {0 = t0 < t1 < t2 < · · · < tn = T }
of [0, T ] set
Π
ft = 2Mti if ti < t ≤ ti+1 , i = 0, 1, 2, · · · , n − 1 and f0Π = 0,
NtΠ = It f Π
and

k−1
X

2

Π

A
=
Mti+1 − Mti
if tk ≤ t < tk+1 , k = 0, 1, 2, · · · , n − 1

 t
i=0
n−1
X

2

Π


A
=
Mti+1 − Mti
 T
i=0
21
.
Then it holds that
2
Mt2 = NtΠ + AΠ
t + (Mt − Mtk )
if tk ≤ t < tk+1 .
(9)
Lemma 29 Let {Mt }t≥0 ∈ M2T and suppose {Mt }t≥0 is bounded, that is, there
exists a C > 0 such that
|Mt (ω)| ≤ C
holds for every t ≥ 0 and ω ∈ Ω.
Then there exists a continuous and non-decreasing process {At }0≤t≤T such that
E
2 n
→0
−
A
sup AΠ
t
t
(10)
0≤t≤T
holds if kΠn k → 0.
Proof. From (6) it follows that
2
2
E NtΠ ≤ 4C 2 E AΠ
+
(M
−
M
)
= 4C 2 E Mt2 ≤ 4C 4 ,
t
tk
t
(11)
E NtΠ − NtΠ
0 2
≤E
sup
0≤s≤t
2 2 Π
Π0
Π∨Π0
fs − fs
At
+ Mt − Mt00k
,
(12)
where Π ∨ Π0 is the partition of [0, T ] generated by Π, Π0 . We easily see from
(9),(11)
2
2
2
E AΠ
+
(M
−
M
)
≤ 2E Mt4 + 2E NtΠ ≤ 10C 4 .
t
t
t
k
Thus (12) implies
E
NtΠ
−
NtΠ
0
2
v
u u
0 4
t
Π
Π
≤ E sup (fs − fs ) E
0≤s≤t
≤
√
0
AtΠ∨Π
s 4
10C
E sup (fsΠ − fsΠ0 ) .
2
+ Mt − Mt00k
2 2
!
(13)
0≤s≤t
4
Since fsΠn converges to {Ms } uniformly on [0, T ] and sup0≤s≤t fsΠn − fsΠm ≤
24 C 4 , the bounded convergence theorem shows the right hand side of (13)
converges to 0 as n, m → n
∞. Then
o Lemma27 concludes that there exists a
Πn
2
{Nt }
∈ MT such that Nt
converges to {Nt }t≥0 on [0, T ]. Therefore
n t≥0
o
n
AΠ
also converges to a continuous stochastic process {At }t≥0 , because
t
t≥0
2
the reminder term (Mt − Mtk ) clearly converges to 0.
Now we can show the decomposition (4).
22
Theorem 30 For {Mt }t≥0 ∈ M2T there exist a unique martingale {Nt }t≥0 and
a non-decreasing stochastic process {At }t≥0 with A0 = 0 such that (4) holds.
Proof. The uniqueness has been proved in Lemma28. The existence of the decomposition is already proved in case M is bounded. If {Mt }t≥0 is not bounded,
we truncate it by
τc = inf {t ≥ 0; |Mt | > c} ,
From (iii) of Theorem18, we see
2
E
sup |Ms |
≤E
sup Ms2 ≤ 4E MT2 ,
0≤s≤T
(14)
0≤s≤T
which assures (3) of Theorem21. Then {Mt∧τc } becomes a bounded continuous
martingale and by the previous argument we have a decomposition
2
Mt∧τ
= Ntc + Act .
c
If c0 > c, then τc0 > τc , hence
0
0
c
Nt∧τ
+ Act∧τc = Ntc + Act .
c
0
0
c
= Ntc , Act∧τc = Act ,
Then the uniqueness of the decomposition implies Nt∧τ
c
therefore we define
Nt = Ntc , At = Act if t ≤ τc .
2
≤ 4E Mt2 , and hence
The property (14) shows E Mt∧τ
c
2
E (At∧τc ) + E M02 = E Mt∧τ
≤ 4E Mt2 .
c
Letting c → ∞, we see At∧τc ↑ At from τc ↑ ∞, hence E (At ) ≤ 4E Mt2 < ∞,
and Nt ∈ L1 . We have to check the martingale property of {Nt } . To verify this
we have only to see the L1 −convergence of Ntc → Nt for every fixed t ≥ 0.
However this is clear from
2
E (|Ntc − Nt |) ≤ E Mt∧τ
− Mt2 + E (|At∧τc − At |)
c
2
= E Mt∧τ
− Mt2 + E (At ) − E(At∧τc ).
c
2
Mt∧τ − Mt2 converges to 0 a.s. as c → ∞ and is dominated by an integrable
c
2 sup0≤s≤T Ms2 . Therefore the first term converges to 0. The second term tends
to 0, because {At } is non-decreasing.
For a given {Mt } ∈ M2T the non-decreasing stochastic process {At }t≥0
in Theorem30 is called the variational process of {Mt }t≥0 and is denoted
by hM it . It is convenient to define a variational process for a product of two
martingales. Making use of an identity
2
2
(Mt + Nt ) − (Mt − Nt )
,
4
for two such processes {Mt , Nt }t≥0 we introduce
Mt N t =
hM + N it − hM − N it
.
4
{hM, N it } can be understood as the unique continuous process of bounded
variation which makes Mt Nt − hM, N it a martingale. This remark implies the
following
hM, N it =
23
Lemma 31 (i) h·, ·it satisfies the property of inner products, that is, h·, ·it is
bilinear and hM, M it ≥ 0.
(ii) For any stopping time τ, Let Mtτ = Mt∧τ , Ntτ = Nt∧τ . Then hM τ , N τ it =
hM, N it∧τ .
From Proposition24 we easily see
Example 32 Let Bt = Bt1 , Bt2 , · · · , Btd t≥0 be a d−dim. B.M. Then B i , B j t =
δij t.
6.2
Stochastic integral
If a process {Yt } is of bounded variation (that is, V1 (Y ) < ∞), an integral
of suitable functions based on the process is possible, which is called as the
Lebesgue-Stiltjes integral. In this section we define a stochastic integral based
on continuous martingales, which was initiated by Ito. As we have seen in the
last section, martingales are not of bounded variation but of bounded variation
of the second order in a weak sence. In this case we have to discuss the integral
keeping in mind the quadratic structure of martingales developed in the last
section.
For stepfunctions we have already defined a stochastic integral (??) based on
martingales. We try to extend the integral to wider class of random functions.
Throughout this section, we fix a large T > 0 and define the integral on [0, T ].
For M ∈ M2T and a function f = (f (t, ω)) on [0, T ] × Ω which is measurable
with respect to B([0, T ]) × F , we define
!
Z
T
2
kf k = E
0
2
|f (t, ω)| d hM it
,
and denote the set of all such functions f satisfying kf k < ∞ by L2 (Ω, P, hM i) .
Clearly this space is a Hilbert space and the space of all stepfunctions L0 restricted on [0, T ] is contained in L2 (Ω, P, hM i) . Define
L2 (hM i) = the closure of L0 in L2 (Ω, P, hM i) .
(15)
The following Lemma tells us a typical function belonging to L2 (hM i) .
Lemma 33 Suppose a function f = (f (t, ω)) is left-continuous for every fixed
ω ∈ Ω and f (t, ·) is Ft −measurable for each t ≥ 0. If f ∈ L2 (Ω, P, hM i) , then
f ∈ L2 (hM i) .
Proof. First suppose f is bounded. For each n ≥ 1, introduce
f (0, ω), if t = 0
fn (t, ω) =
.
f 2kn , ω , if t ∈ 2kn , k+1
2n , k = 0, 1, 2, · · ·
Then fn ∈ L0 and the left-continuity implies fn (t, ω) → f (t, ω) for any fixed
(t, ω) ∈ [0, T ]×Ω. Therefore the bounded convergence theorem shows kfn → f k →
0, hence f ∈ L2 (hM i) . If f is not bounded, we prepare

 N, if x ≥ N
x,
if |x| < N ,
ϕN (x) =

−N, if x ≤ −N
24
and set f N = ϕN (f ). Then f N ∈ L2 (hM i) and it is easy to see that f N → f →
0 as N → ∞, hence f ∈ L2 (M ) .
Now we look back (6). Since Mt2 = Nt + hM it , we have
E ft2i ∆2i Fs = ft2i E ∆2i Fs = ft2i hM iti+1 − hM iti .
Therefore we see
2
2
E It (f ) |Fs − Is (f ) = E
k−1
X
ft2i ∆2i
+
!
ft2k ∆2t Fs
fu (ω)2 d hM iu Fs
i=l
Z t
=E
s
In particular, setting s = 0 and taking expectation, we have
Z t
fu (ω)2 d hM iu .
E It (f )2 = E
(16)
(17)
0
Notice I· (f ) ∈ M2T and M2T is a Hilbert space (Lemma27). For f ∈ L2 (M ) ,
choose a sequence of {fn } ⊂ L0 such that kf − fn k → 0 as n → ∞. Then (17)
shows
!
Z T
2
2
2
E (IT (fn ) − IT (fm )) = E
(fn,u (ω) − fm,u (ω)) d hM iu = kfn − fm k ,
0
hence {I· (fn )}n≥1 converges in M2T , and its limit is denoted by I· (f ) (= I·M (f )
if necessary). This I·M (f ) is called a stochastic integral based on a martingale
M. Now suppose M, N ∈ M2T and f ∈ L2 (hM i) , g ∈ L2 (hN i) . Then I·M (f )
and I·N (g) are in M2T . The next problem is to compute their h·, ·it . For this
purpose, first we assume f, g ∈ L0 . Then a similar calculation as (6),(15) shows
!
k−1
X
E ItM (f )ItN (g) |Fs − IsM (f )IsN (g) = E
fti gti ∆i ∆0i + ftk gtk ∆t ∆0t Fs
i=l
Z t
=E
fu (ω)gu (ω)d hM, N iu Fs
s
with
∆0i
= Nti+1 − Nti ,
∆0t
= Nt − Ntk . Therefore we have
Z t
M
I· (f ), I·N (g) t =
fu (ω)gu (ω)d hM, N iu
(18)
0
Since the convergence of the right hand side of (18) is not trivial for general
f ∈ L2 (hM i) , g ∈ L2 (hN i) , we have to examine it. From (18) it follows that
for f, g ∈ L0
Z
T
q
fu (ω)gu (ω)d hM, N iu ≤ hI·M (f )iT hI·N (g)iT
0
s
s
Z T
Z T
=
fu (ω)2 d hM iu
gu (ω)2 d hN iu . (19)
0
0
25
Let h(s) be a non-random stepfunction. Then for any bounded variation function v(t), we see
Z
Z
T
T
d |v| (u),
h(u)dv(u) =
sup khk∞ ≤1 0
0
where khk∞ = sup {|h(u)| ; u ∈ [0, T ]} and |v| (u) is the total variation on [0, u]
of v. Let h1 , h2 be two non-random stepfunctions. Then in (19) replacing f, g
by h1 f, h2 g respectively, we have
s
s
Z T
Z T
Z T
2
|fu (ω)gu (ω)| d |hM, N i|u ≤
fu (ω) d hM iu
gu (ω)2 d hN iu . (20)
0
0
0
Now it is easy to see that (20) holds for any f ∈ L2 (hM i) , g ∈ L2 (hN i) .
Summing up these argument, we obtain
Theorem 34 Suppose M, N ∈ M2T and f ∈ L2 (hM i) , g ∈ L2 (hN i) . Then
(20) and (18) are valid .
We remark the following
Lemma 35 Let M ∈ M2T , f ∈ L2 (hM i) and.τ be a stopping time. Then
τ
M
(f ) = ItM (f τ ), where Mtτ = Mt∧τ , ftτ = I[0,τ ] (t)ft .
It∧τ
Proof. For f ∈ L0 , the statement is clearly valid and then pass to the limit. In
the calculation notice (ii) of Lemma31.
6.3
Localization of stochastic integral
It is not convenient that the stochastic integral is defined only for square integrable martingales. In this section we try to extend the notion of martingales
with the help of the optional stopping theorem. Set
M = {Mt }0≤t≤T ; there exists stopping times {τn }n≥1 such
,
M2loc,T =
that 0 ≤ τn ≤ τn+1 ↑ ∞ and M τn = {Mt∧τn }0≤t≤T ∈ M2T
and
L2loc (hM i)
f = {ft }0≤t≤T ; there exists stopping
times{τn }n≥1 such that
=
.
0 ≤ τn ≤ τn+1 ↑ ∞ and f τn = I[0,τn ] (t)ft 0≤t≤T ∈ L2 (hM i)
An element of M2loc,T is called a local martingale. For M ∈ M2loc,T , f ∈
τn
L2loc (hM i) , ItM ( f τn ) is defined as stochastic integrals introduced in Section5.2. If n > m, then from Lemma35 we have
ItM
τm
( f τm ) = ItM
τn ∧τm
τn
M
( f τn ∧τm ) = It∧τ
( f τn ) .
m
Therefore we can define
ItM (f ) = ItM
τn
( f τn ) if t ≤ τn .
26
It is clear that I·M ( f ) ∈ M2loc,T . For M, N ∈ M2loc,T , we define
hM, N it = hM τn , N τn it if t ≤ τn .
This definition can be seen well-defined through (ii) of Lemma31. It is easy to
see that
Z t
M
I· (f ) , I·N (g) t =
fs gs d hM, N is ,
(21)
0
for M, N ∈
6.4
M2loc,T
and f, g ∈
L2loc
(hM i) .
Ito’s formula
If At is a continuous function with bounded variation (of the first order) and f
is a differentiable function whose derivative f 0 is continuous, then the following
formula is valid.
Z
t
f 0 (As )dAs .
f (At ) − f (A0 ) =
(22)
0
This can be seen as follows. Let Π = {0 = t0 < t1 < t2 < · · · < tn = t} . Set
f (y) − f (x) = f 0 (x)(y − x) + ε1 (x, y)(y − x).
Then for a large enough C such that As ∈ [−C, C] for any s ∈ [0, t]
ρ1 (ε) ≡ sup {|ε1 (x, y)| ; |y − x| ≤ ε, x, y ∈ [−C, C] } → 0
as
ε → 0.
Thus
f (At ) − f (A0 ) =
n−1
X
f (Atk+1 ) − f (Atk )
k=0
=
n−1
X
X
n−1
f 0 (Atk ) Atk+1 − Atk +
ε1 Atk , Atk+1 Atk+1 − Atk
k=0
k=0
Then as kΠk → 0,
n−1
X
0
Z
f (Atk ) Atk+1 − Atk →
t
f 0 (As )dAs ,
0
k=0
and
n−1
X
X
n−1
ε1 At , At
At
ε1 Atk , Atk+1 Atk+1 − Atk ≤
− Atk k
k+1
k+1
k=0
k=0
≤ ρ1 (kΠk) V1 (A) → 0.
If the process At is replaced by a martingale Mt , the formula corresponding to
(22) is no longer valid, and we have to taking accout of the next term of the
Taylor expansion of f.
27
Theorem 36 Let M i = Mti ∈ M2loc,T and Ait be adapted continuous processes with bounded variation for 1 ≤ i ≤ d. Then for f = f (x1 , x2 , · · · , xd )
which has continuous derivatives up to the second order with respect to any
variables, we have
d Z
X
f (Xt ) − f (X0 ) =
d
t
fxi (Xs )dXsi
Z
t
i,j=! 0
0
i=!
1X
+
2
fxi ,xj (Xs ) d M i , M j s , (23)
where Xt = Xt1 , Xt2 , · · · , Xtd and Xti = Mti + Ait .
Proof. For simplicity we assume d = 1. We assume first M, A are bounded by
C. From the assumption on f it follows that
1
2
f (y) − f (x) = f 0 (x)(y − x) + f 00 (x)(y − x)2 + ε2 (x, y) |y − x| ,
2
with
ρ2 (ε) ≡ sup {|ε2 (x, y)| ; |y − x| ≤ ε, |x| , |y| ≤ 2C } → 0
as
ε → 0.
Then for Π = {0 = t0 < t1 < t2 < · · · < tn = t} we see
f (Xt ) − f (X0 ) =
n−1
X
X
n−1
f (Xtk+1 ) − f (Xtk ) =
f 0 (Xtk ) Xtk+1 − Xtk
k=0
+
1
2
n−1
X
k=0
f 00 (Xtk ) Xtk+1 − Xtk
2
k=0
+
n−1
X
2
ε2 Xtk , Xtk+1 Xtk+1 − Xtk k=0
≡ I1Π + I2Π + I3Π .
Since f 0 (Xt ) is continuous and bounded, it is clear that
n−1
X
I1Π =
X
n−1
f 0 (Xtk ) Mtk+1 − Mtk +
f 0 (Xtk ) Atk+1 − Atk
k=0
Z t
→
k=0
0
Z
f (Xs )dMs +
0
t
Z
0
f (Xs )dAs =
0
t
f 0 (Xs )dXs ,
0
as kΠk → 0. Here the first term converges in L2 (Ω, P ) from Theorem34. To
treat I3Π , set
ε (Π, X) =
sup
|Xt − Xs | .
0≤s,t≤T, |s−t|≤kΠk
Then
n−1
X
Π
I3 ≤ ρ2 (ε (Π, X))
Xt
k+1
2
− Xtk .
k=0
However
n−1
X
X t
k+1
n−1
n−1
X
X
2
2
2
At
+2
Mt
− Xtk ≤ 2
−
A
− Mtk t
k+1
k
k+1
k=0
k=0
and
n−1
X
At
k+1
k=0
2
− Atk ≤ ε (Π, A) V1 (A) → 0 as kΠk → 0.
k=0
28
(24)
The second term is nothing but
n−1
X
Mt
k+1
2
− Mtk = AΠ
t (see(9)).
k=0
n
Then from Lemma29 it follows that AΠ
→ hM it in L2 (Ω, P ) as kΠn k → 0.
t
Πn
Hence, by choosing a suitable subsequence {nk } , we see At k → hM it uniformly
Πnk
on [0, T ] a.s.. Then (24) shows I3
→ 0. As for I2Π , only the term
J2Π ≡
n−1
X
f 00 (Xtk ) Mtk+1 − Mtk
2
k=0
has a non-trivial limit. However we have
Z t
Π
J2 =
f 00 (Xs )dAΠ
s.
0
Πn
At k
Πn
As we have seen that
→ hM it uniformly on [0, T ] a.s., hence J2 k →
Z t
f 00 (Xs )d hM is . Consequently we have proved (23) if d = 1 and M is bounded.
0
For a general M ∈ M2loc,T , we have only to truncate it, namely Mtτn ∧τc =
0
Mt∧τn ∧τc . Set τ 0 = τn ∧ τc . Then (23) is valid for M τ , hence from (ii) of
Lemma31 and the definitions in Section5.3 it follows that
Z
Z t
D
E
0 2
1 t 00
0
f (Xs∧τ 0 )d M τ
f (Xt∧τ 0 ) − f (X0 ) =
f (Xs∧τ 0 )dXs∧τ 0 +
2 0
s
0
Z t∧τ 0
Z t∧τ 0
1
2
=
f 0 (Xs )dXs +
f 00 (Xs )d hM is .
2
0
0
Since τ 0 = τn ∧ τc ↑ ∞ as n, c → ∞ we complete the proof if d = 1.
A process Xt = Mt + At with a continuous local martingale M and a continuous adapted process with bounded variation is called a semi-martingale.
Ito’s formula is a chain rule in stochastic analysis and it can be understood more
intuitively if we write (23) in a differential form:
df (Xt ) =
d
X
d
fxi (Xt )dXti +
i=!
1X
fxi ,xj (Xt ) d M i , M j t .
2
i,j=!
The variational process is expressed sometimes as
d hM, N it = dMt dNt ,
and if A, A0 are continuous processes with bounded variation, we set
dAt dMt = dMt dAt = dAt dA0t = 0.
Then the Ito’s formula can be written as
df (Xt ) =
d
X
d
fxi (Xt )dXti +
i=!
1X
fxi ,xj (Xt ) dX i dXtj .
2
i,j=!
29
7
Applications
Ito’s formula has many applications, however here we give two of them. For
other applications see the exercises.
7.1
Martingale characterization of Brownian motions
Lévy pointed out that Brownian motions can be characterized by martingales
and it led Stroock-Varadhan to characterization of various stochastic processes
through martingales, which is now called the martingale problem.
Theorem 37 Let M i ∈ M2loc,T with M i , M j t = δij t and M0i = 0 for i =
1, 2, · · · , d. Then Mt = Mt1 , Mt2 , · · · , Mtd is a d−dimensional Brownian motion.
Proof. For ξ ∈ Rd apply Ito’s formula to eiξ·Mt :
iξ·Mt
e
Z t
Z t
d
d
X
1X
iξ·Ms
k
ξk ξl eiξ·Ms d M k , M l s
= 1 + i ξk e
dMs −
2
0
0
k,l=1
k=1
Z t
Z t
d
X
1
2
iξ·Ms
k
eiξ·Ms ds.
= 1 + i ξk e
dMs − kξk
2
0
0
k=1
Thus we have for t ≥ s ≥ 0
1
2
E eiξ·Mt |Fs = eiξ·Ms − kξk
2
Z
t
E eiξ·Mu |Fs du.
s
Regarding this relationship as an integral equation with unknown E eiξ·Mt |Fs ,
we obtain
2
1
E eiξ·Mt |Fs = eiξ·Ms e− 2 (t−s)kξk ,
which is equivalent to
2
1
E eiξ·(Mt −Ms ) |Fs = e− 2 (t−s)kξk .
This shows that Mt − Ms is independent of Fs and Mt − Ms is distributed as
a Gaussian distribution of expectation 0 and variance matrix (t − s) I, which
concludes that Mt is a d−dim.B.M..
Corollary 38 Let Bt be a d−dim.B.M. and (gij (t, ω))1≤i,j≤d be an orthogo2
nal matrix for every
t ≥ 0 and ω ∈ Ω. Assume gij ∈ L (hBi) . Then Mt =
1
2
d
Mt , Mt , · · · , Mt with
Mti =
d Z
X
t
gij (s, ω)dBsj
j=1 0
becomes a d−dim.B.M..
30
7.2
Brownian functionals and stochastic integral
Let Bt = Bt1 , Bt2 , · · · , Btd be a d−dim.B.M. starting from 0 and set
Ft = σ − {Bs ; 0 ≤ s ≤ t} = σ − {Bs − Bu ; 0 ≤ u ≤ s ≤ t} .
In this section we show that any FT −measurable square integrable random
variable can be expressed by a stochastic integral based on {Bt } . For simplicity
assume d = 1. Set


m

 linear combination of Y eiξk (Brk −Brk−1 ) 

F=
.
k=1




with ξk ∈ R, 0 ≤ r0 ≤ r1 ≤ · · · ≤ rm ≤ T
Lemma 39 F is dense in L2 (Ω, FT , P ) .
Proof. Let
Y = f Br1 − Br0 , · · · , Brm − Brm−1 ; f is a Borel measurable
F1 =
.
2
function on Rm with E f Br1 − Br0 , · · · , Brm − Brm−1 < ∞
Then
F ⊂ F1 and F1 is dense in L2 (Ω, FT ,P ) . Hence we have only to prove that
linear combination of eiξ·x with ξ ∈ Rm is dense in L2 (Rm , N (0, Λ)), where
N (0, Λ) is a Gaussian distribution with 0 expectation and covariance matrix


r1 − r 0
0
···
0


0
r 2 − r1 · · ·
0



.
..
..
..
..


.
.
.
.
0
···
0
rm − rm−1
However this is clear by the Fourier transform.
Theorem 40 For any X ∈ L2 (Ω, FT , P ) there exists a unique f ∈ L2 (hBi)
such that
Z
T
X = EX +
f (s, ω) dBs .
(25)
0
Proof. Let S be the set of all X which is represented as (25).
1) X = eiξ(Bt −Br ) ∈ S ,(0 ≤ r ≤ t ≤ T ).
Due to Ito’s formula we have
Z T
1 2
1 2
I[r,t) (s)iξeiξ(Bs −Br ) e 2 ξ (s−t) dBs ,
X = e− 2 ξ (t−r) +
r
1
2
hence setting f (s, ω) = I[r,t) (s)iξeiξ(Bs −Br ) e 2 ξ (s−t) , we obtain eiξ(Bt −Br ) ∈ S.
2) Suppose Xk ∈ S with bounded fk for k = 1, 2, · · · , n and fk (s, ω)fl (s, ω) = 0
for k 6= l. Then X1 X2 · · · Xn ∈ S.
Let
Z
t
Xk (t) = EXk +
fk (s, ω)dBs .
0
31
Then Ito’s formula implies
X1 (t)X2 (t)
t
Z
t
Z
= X1 (0)X2 (0) +
X1 (s)dX2 (s) +
0
Z
0
Z
= (EX1 ) (EX2 ) +
t
d hX1 , X2 is
X2 (s)dX1 (s) +
0
t
Z
(X1 (s)f2 (s) + X2 (s)f1 (s)) dBs +
0
t
f1 (s)f2 (s)ds,
0
thus
Z
T
(X1 (s)f2 (s) + X2 (s)f1 (s)) dBs ∈ S.
X1 X2 = (EX1 ) (EX2 ) +
0
Inductively we can show X1 X2 · · · Xn ∈ S.
3) Suppose Xn ∈ S and Xn → X in L2 (Ω, P ). Then X ∈ S.
This follows immediately from Lemma27.
Summing up the above argument and Lemma39 conclude the theorem.
For higher dimensional Brownian motions also the corresponding theorem is
valid, that is,
Theorem 41 Let (Ω, FT , P ) be a probability space generated by a d−dim.B.M.
{Bt } . Then for any X ∈ L2 (Ω, FT , P ) ,there exist unique fi ∈ L2 B i
(i = 1, 2, · · · , d) such that
X = EX +
d Z
X
T
fi (s, ω) dBsi .
i=1 0
This theorem was first proved by K.Ito by applying the Wiener-Ito expansion
and is now used to Black-Sholes theory to show the completeness of markets.
Corollary 42 Let (Ω, FT , P ) be a probability space generated by a d−dim.B.M.
{Bt } . Then Ft+0 = Ft for every t ≥ 0.
Proof. Let X ∈ L2 (Ω, Ft+0 , P ) . Then for any n ≥ 1, X ∈ L2 Ω, Ft+1/n , P ,
and applying Theorem41, we have
X = EX +
d Z
X
t+1/n
fin (s, ω) dBsi .
i=1 0
However the uniqueness implies
fin (s, ω) = fim (s, ω) if s ∈ [0, t +
1
] and n ≥ m,
n
hence we see
X = EX +
d Z
X
i=1 0
32
t
fi1 (s, ω) dBsi .
Corollary 43 Let (Ω, FT , P ) be a probability space generated by a d−dim. B.M.
{Bt } . Any square integrable martingale M = {Mt } on this probability space
can be represented by a stochastic integral based on the Brownian motion, and
hence is automatically continuous.
Proof. Since MT ∈ L2 (Ω, FT , P ) , Theorem41 implies
MT = EMT +
d Z
X
T
fi (s, ω) dBsi .
i=1 0
Therefore we see
d Z
X
Mt = E (MT |Ft ) = EMT +
t
fi (s, ω) dBsi .
i=1 0
8
Stochastic differential equations
Ito constructed stochastic integral based on Brownian motions in order to define
rigorously (ordinary) differential equations whose coefficients are derivatives of
Brownian motions. His equatios are now called as stochastic differential equations (SDE) and are used to describe various random phenomena. This section
is devoted to a brief introduction to the theory of SDE.
Let {aij (t, x), bi (t, x)}1≤i≤d,1≤j≤d0 be functions on [0, ∞) × Rd satisfying
certain smoothness conditions. A SDE is
0
dXti
=
d
X
aij (t, Xt ) dBtj + bi (t, Xt ) dt,
1 ≤ i ≤ d.
j=1
More precisely this equation has to be understood in an integral form:
0
Xti
= xi +
d Z
X
t
aij (s, Xs ) dBsj +
j=1 0
Z
t
bi (s, Xs ) ds,
1 ≤ i ≤ d.
(26)
0
If aij (t, x) = 0, then the above equation becomes an ordinary differential equation. We prove here the existence and uniqueness of the equation (26). Assume
{aij (t, x), bi (t, x)}1≤i≤d,1≤j≤d0 are continuous functions [0, T ] × Rd satisfying
|aij (t, x) − aij (t, y)| + |bi (t, x) − bi (t, y)| ≤ K kx − yk
(27)
with some K > 0 and for any x, y ∈ Rd , t ∈ [0, T ]. Set
n
o
d
L2c = Xt = Xti (ω) i=1 ; Xt is continuous,adapted and satisfying kXkT < ∞ ,
where
s 2
kXkt = E sup kXs k .
0≤s≤t
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Theorem 44 Under the condition (27) the equation (26) has a unique solution
for any initial condition (xi )1≤i≤d ∈ Rd .
Proof. For {Xt } ∈ L2c define
0
i
Φ (X)t
= xi +
d Z
X
t
aij (s, Xs ) dBsj +
t
Z
1 ≤ i ≤ d.
bi (s, Xs ) ds,
0
j=1 0
Since
|aij (t, x)| + |bi (t, x)| ≤ K kxk + |aij (t, 0)| + |bi (t, 0)| ≤ K kxk + K1
(28)
with K1 = max {|aij (t, 0)| + |bi (t, 0)| ; 0 ≤ t ≤ T, 1 ≤ i ≤ d, 1 ≤ j ≤ d0 } , we see
from (iii) of Theorem18 and (28)
2 i
E sup Φ (X)t 0≤t≤T

2 
0
2 !
Z t
d Z t
X


j
2
aij (s, Xs ) dBs  + 2E
sup bi (s, Xs ) ds
≤ 2xi + 2E  sup 0≤t≤T
0≤t≤T j=1 0
0
 0

!
Z T
d Z T
X
2
2
2


≤ 2xi + 8E
|aij (s, Xs )| ds + 2T E
|bi (s, Xs )| ds
j=1 0
≤ 2x2i + 16K 2 d0 T E
0
2
sup kXt k
2
sup kXt k
+ 4T E
0≤t≤T
+ K2
0≤t≤T
with some other constant K2 . Therefore Φ (X) ∈ L2c . Similarly from (27) we
have for X, Y ∈L2c
Z t
2
2
kΦ (X) − Φ (Y)kt ≤ K3
kX − Yks ds
(29)
0
with some constant K3 . Therefore setting
X0t = (xi ) , Xnt = Φ Xn−1
we see
n+1
2
X
− Xn t ≤ K3
Z
0
t
t
for n ≥ 1,
n
X − Xn−1 2 ds for n ≥ 1,
s
which implies
n
n+1
2
(K3 t)
X
− Xn t ≤
n!
Thus
Z
0
t
1
X − X0 2 ds.
s
∞
X
n+1
X
− Xn T < ∞.
n=0
L2c
n
Since
is complete as we have seen in Lemma27, there exists an X ∈L2c such
that X → X in L2c . This X satisfies X =Φ (X) , hence (26) . The uniqueness
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follows from a truncation argument. Let {Xt , X0t } be two solutions of (26) and
set
τc = inf {t ≥ 0; kXt k ∨ kX0t k ≥ c} .
Then we see
2 E Xt∧τc − X0t∧τc ≤ K3
hence
t
Z
0
2 E Xs∧τc − X0s∧τc ds,
2 E Xt∧τc − X0t∧τc = 0,
which implies the uniqueness.
8.1
SDE and partial differential equations
Let {Bt } be a d−dim.B.M. and u(t, x) be a smooth bounded function on [0, ∞)×
Rd satisfying
∂u
1
= ∆u, u(0, x) = f (x).
(30)
∂t
2
Then applying Ito’s formula to Mt = u(T − t, Bt + x), we see
X
dMt = −ut (T − t, Bt + x)dt +
uxi (T − t, Bt + x) dBti
1≤i≤d
1 X
+
uxi ,xj (T − t, Bt + x) dBti dBtj
2
1≤i,j≤d
X
= −ut (T − t, Bt + x)dt +
uxi (T − t, Bt + x) dBti
1≤i≤d
1
+ ∆u (T − t, Bt + x) dt
2
X
=
uxi (T − t, Bt + x) dBti ,
1≤i≤d
hence {Mt } turns out to be a martingale. Therefore we have
EMT = EM0 =⇒ u(T, x) = Ef (BT + x).
This shows the uniqueness of the bounded solutions for the equation (30) and
u(t, x) = Ef (Bt + x)
satisfies the equation (30). This argument can be extended to the solutions of
SDE whose coefficients do not depend on t. Under the condition of Theorem44
on the coefficients, set
L=
X
1 X
∂2
∂
σij (x)
+
bi (x)
,
2
∂xi ∂xj
∂xi
1≤i,j≤d
1≤i≤d
where
σij (x) =
X
1≤k≤d0
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aik (x)ajk (x).
Theorem 45 (Feynman-Kac formula) Suppose u(t, x) are bounded, differentiable w.r.t. t and twice differentiable w.r.t. x. Assume V (t, x) is a measurable
function bounded from below and u satisfies
∂u
= Lu − V u, u(0, x) = f (x).
∂t
Assume {Xt } is the solution of (26).Then
Z t
u(t, x) = E f (Xt ) exp − V (t − s, Xs )ds
.
0
Proof. Setting
Z t
Mt = u(T − t, Xt ) exp − V (T − s, Xs )ds ,
0
we have only to apply Ito’s formula to {Mt } .
Apart from this theorem SDEs are related to many kinds of partial differential equations and partial differential equations can be studied through SDEs,
and vice versa.
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