1. Stochastic Processes and filtrations
1. Stoch. pr.,
filtrations
2. BM as
weak limit
A stochastic process (Xt )t∈T is a collection of random variables on
(Ω, F) with values in a measurable space (S, S), i.e., for all t,
Xt : Ω → S is F − S-measurable.
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
In our case
1
T = {0, 1, . . . , N} or T = N ∪ {0} (discrete time)
2
or T = [0, ∞) (continuous time).
The state space will be (S, S) = (R, B(R)) (or (S, S) = (Rd , B(Rd ))
for some d ≥ 2).
Example 1.1
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Random walk:
Let εn , n ≥ 1, be a sequence of iid random variables with
(
1
with probability 21
εn =
−1 with probability 12
Define X̃0 = 0 and, for k = 1, 2, . . . ,
X̃k =
k
X
i=1
εi
1. Stoch. pr.,
filtrations
2. BM as
weak limit
For a fixed ω ∈ Ω the function
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
t 7→ Xt (ω)
is called (sample) path (or realization) associated to ω of the
stochastic process X .
Let X and Y be two stochastic processes on (Ω, F).
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Definition 1.2
(i) X and Y have the same finite-dimensional distributions if, for
all n ≥ 1, for all t1 , . . . , tn ∈ [0, ∞) and A ∈ B(Rn ):
P((Xt1 , Xt2 , . . . , Xtn ) ∈ A) = P((Yt1 , Yt2 , . . . , Ytn ) ∈ A).
(ii) X and Y are modifications of each other, if, for each t ∈ [0, ∞),
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
P(Xt = Yt ) = 1.
(iii) X and Y are indistinguishable, if
P(Xt = Yt for every t ∈ [0, ∞)) = 1.
Example 1.3
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1. Stoch. pr.,
filtrations
Definition 1.4
A stochastic process X is called continuous (right continuous), if
almost all paths are continuous (right continuous), i.e.,
P({ω : t 7→ Xt (ω) continuous (right continuous)}) = 1
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
.
Proposition 1.5
If X and Y are modifications of each other and if both processes are
a.s. right continuous then X and Y are indistinguishable.
3. Gaussian p.
Let (Ω, F) be given and {Ft , t ≥ 0} be an increasing family of
σ-algebras (i.e., for s < t, Fs ⊆ Ft ), such that Ft ⊆ F for all t.
That’s a
filtration.
4. Stopping
times
For example:
5. Cond.
expectation
Definition 1.6
The filtration FtX generated by a stochastic process X is given by
1. Stoch. pr.,
filtrations
2. BM as
weak limit
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
FtX = σ{Xs , s ≤ t}.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 1.7
A stochastic process is adapted to a filtration (Ft )t≥0 if, for each
t≥0
Xt : (Ω, Ft ) → (R, B(R))
is measurable. (Short: Xt is Ft -measurable.)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 1.8 (”The usual conditions”)
Let Ft + = ∩ε>0 Ft+ε . We say that a filtration (Ft )t≥0 satisfies the
usual conditions if
1
(Ft )t≥0 is right-continuous, that is, Ft + = Ft , for all t ≥ 0.
2
(Ft )t≥0 is complete. That is, F0 contains all subsets of
P-nullsets.
2. Brownian motion as a weak limit of random
walks
Example 2.1
Scaled random walk: for tk =
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
X̃tNk
k
N,
k = 0, 1, 2, . . . , we set
k
1
1 X
√
√
:=
εi
X̃k =
N
N i=1
Definition 2.2 (Brownian motion)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
A standard, one-dimensional Brownian motion is a continuous
adapted process W on (Ω, F, (Ft )t≥0 , P) with W0 = 0 a.s. such
that, for all 0 ≤ s < t,
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1
Wt − Ws is independent of Fs
2
Wt − Ws is normally distributed with mean 0 and variance t − s.
Recall the scaled random walk: for tk =
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
k
N,
k = 0, 1, 2, . . . , we set
k
1
1 X
εi
X̃tNk := √ X̃k = √
N
N i=1
By interpolation we define a continuous process on [0, ∞):


[Nt]
X
1
XtN = √ 
εi + (Nt − [Nt])ε[Nt]+1 
N i=1
(2.1)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 2.3 (Donsker)
Let (Ω, F, P) be a probability space with a sequence of iid random
variables with mean 0 and variance 1. Define (XtN )t≥0 as in (2.1).
Then (XtN )t≥0 converges to (Wt )t≥0 weakly.
Clarification of the statement:
1. Stoch. pr.,
filtrations
1
Ω̃ = C [0, ∞)
2
On Ω̃ = C [0, ∞) one can construct a probability measure P ∗
(the Wiener measure) such that the canonical process
(Wt )t≥0 is a Brownian motion. Let ω = ω(t)t≥0 ∈ C [0, ∞).
Then the canonical process is given as Wt (ω) := ω(t).
3
(XtN )t≥0 defines a measure probability P N on Ω̃ = C [0, ∞).
Indeed,
X N : (Ω, F) → (C [0, ∞), B(C [0, ∞))
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
measurable, that means we consider the random variable
ω 7→ X N (ω, ·), which maps ω to a continous function in t (i.e.
the path). Let A ∈ B(C [0, ∞)), then
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
P N (A) := P({ω : X N (ω, · ∈ A}).
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
4
So, Donsker’s theorem says that on
w
(Ω̃, F̃) = (C [0, ∞), B(C [0, ∞))) we have that P N −
→ P ∗.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Steps of the proof:
1
3. Gaussian p.
4. Stopping
times
w
→ (Ws1 , Ws2 , . . . , Wsn ).
(XsN1 , XsN2 , . . . , XsNn ) −
5. Cond.
expectation
6. Martingales
2
Show that (P N )N≥1 is a tight familiy of probability measures on
(Ω̃, F̃).
3
(1) and (2) imply P N −
→ P ∗.
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Show that the finite-dimensional distributions of X N converge
to the finite-dimensional distributions of W . That means, show
that, for all n ≥ 1, and all s1 , . . . , sn ∈ [0, ∞)
w
Easy partial result of step (1) of the proof:
By (CLT), for fixed t ∈ [0, ∞), for N → ∞,
1. Stoch. pr.,
filtrations
w
XtN −
→ Wt
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
Theorem 2.4 (CLT)
Let (ξn )n≥1 be iid, mean=0, variance=σ 2 . Then, for each x ∈ R,
!
n
1 X
lim P √
ξi ≤ x = Φσ (x),
n→∞
n i=1
where Φσ (x) is the distribution function of N(0, σ 2 ). In other words
8. Refl. princ.,
pass.times
n
1 X w
√
ξi −
→ Z,
n i=1
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
where Z ∼ N(0, σ 2 ).
3. Brownian motion as a Gaussian process
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
Definition 3.1
A stochastic process (Xt )t≥0 is called Gaussian process, if for each
finite familiy of time points {t1 , . . . , tn } the vector (Xt1 , . . . , Xtn ) has
a multivariate normal distribution.
A Gaussian process is called centered if E [Xt ] = 0 for all t.
The covariance function is given by γ(s, t) = Cov(Xs , Xt ).
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 3.2
A Brownian motion is a centered Gaussian process with
γ(s, t) = s ∧ t (= min(s, t)). Conversely, a centered Gaussian process
with continuous paths and covariance function γ(s, t) = s ∧ t is a
Brownian motion.
4. Stopping times
Given is a filtered probability space (Ω, F, (Ft )t≥0 , P).
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
A random time T is a measurable function
T : (Ω, F) → ([0, ∞], B([0, ∞]).
Definition 4.1
If X is a stochastic process and T a random time, we define the
function XT on {T < ∞} by:
XT (ω) = XT (ω) (ω)
Definition 4.2
A random time T is a stopping time for (Ft )t≥0 if, for all t ≥ 0,
1. Stoch. pr.,
filtrations
{T ≤ t} = {ω : T (ω) ≤ t} ∈ Ft .
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Proposition 4.3
1
If T (ω) = t a.s. for some constant t ≥ 0, then T is a stopping
time.
2
Every stopping time T satisfies that, for all t,
6. Martingales
7. Discrete
stoch. integral
{T < t} ∈ Ft .
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
3
If the filtration is right continuous and a random time T
satisfies (4.1) for all t, then T is a stopping time.
(4.1)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
Lemma 4.4
Let S and T be stopping times for (Ft )t≥0 . Then S ∧ T , S ∨ T and
S + T are stopping times for (Ft )t≥0 .
Definition 4.5
Let T be a stopping time for (Ft )t≥0 . The σ-field FT of events
determined prior to the stopping time T is given by
5. Cond.
expectation
FT = {A ∈ F : A ∩ {T ≤ t} ∈ Ft , for all t ≥ 0}.
6. Martingales
7. Discrete
stoch. integral
Remark:
8. Refl. princ.,
pass.times
1
FT is a σ-field.
9. Quad.var.,
path prop.
2
T is FT -measurable.
3
If T (ω) = t a.s., then FT = Ft .
10. Ito integral
11. Ito’s
formula
Example 4.6
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Let (Wt )t≥0 be a standard Brownian motion on (Ω, F, (Ft )t≥0 , P).
Let Ta be the first passage time of the level a ∈ R, i.e.,
Ta (ω) = inf{t > 0 : Wt (ω) = a}.
Then Ta is a stopping time for (Ft )t≥0 .
(4.2)
Remark: Let S and T be stopping times for (Ft )t≥0 .
1. Stoch. pr.,
filtrations
1
For A ∈ FS we have that A ∩ {S ≤ T } ∈ FT .
2
If, moreover, S ≤ T then FS ⊆ FT .
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 4.7
A stochastic process (Xt )t≥0 is called progressively measurable for
(Ft )t≥0 if, for all t and all A ∈ B(R), the set
{(s, ω) : 0 ≤ s ≤ t, ω ∈ Ω, Xs (ω) ∈ A} ∈ B([0, t]) ⊗ Ft ,
that means
(s, ω) 7→ Xs (ω) is B([0, t]) ⊗ Ft − B(R) − measurable.
1. Stoch. pr.,
filtrations
Example: If (Xt )t≥0 is adapted and right-continuous then it is
progressively measurable.
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
Proposition 4.8
5. Cond.
expectation
Let X be progressively measurable on (Ω, F, (Ft )t≥0 ) and let T be a
stopping time for (Ft )t≥0 . Then
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1
XT defined on {T < ∞} is an FT -measurable random variable
and
2
the stopped process (XT ∧t )t≥0 is progressively measurable.
5. Conditional expected value
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Definition 5.1
Let (Ω, F, P) be a probability space and X : Ω → R be an
F − B(R)-measurable
random variable such that
R
E [|X |] = |X |dP < ∞. Let G ⊆ F be a (smaller) σ-algebra.
Then there exists a random variable Y with the following properties:
1
Y is G − B(R)-measurable.
2
Y is integrable, i.e., E [|Y |] < ∞.
3
For all A ∈ G:
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
E [Y 1IA ] = E [X 1IA ].
The G-measurable random variable Y is called conditional expected
value of X given G. We write Y = E [X |G] a.s.
Remark:
1. Stoch. pr.,
filtrations
1
E [X |G] is unique with respect to equality a.s.
2
Suppose that, additionally, E [X 2 ] < ∞. Then E [X |G] is the
orthogonal projection of X ∈ L2 (Ω, F) onto the subspace
L2 (Ω, G).
3
If G = {∅, Ω} then E [X |G] = E [X ] a.s.
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 5.2
Let Z be a function on Ω. Then E [X |Z ] := E [X |σ(Z )].
Example 5.3
Example 5.4
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
S∞
X : (Ω, F) → (R, B(R)). Suppose Ω = k=1 Ωk such that
Ωi ∩ Ωj = ∅ for i 6= j. Let G := σ{Ωk , k ≥ 1}.
E [X |G] =
Example 5.5
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Suppose (X , Z ) has a bivariate density f (x, z).
Theorem 5.6 (A list of properties of conditional expectation)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
In the following let X , Y , Xn , n ≥ 1, be integrable random variables
on (Ω, F, P). The following holds
(a) If X is G-measurable, then E [X |G] = X a.s.
3. Gaussian p.
4. Stopping
times
(b) Linearity: E [aX + bY |G] = aE [X |G] + bE [Y |G] a.s.
5. Cond.
expectation
6. Martingales
(c) Positivity: If X ≥ 0 a.s. then E [X |G] ≥ 0 a.s.
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
(d) Monotone convergence (MON): If 0 ≤ Xn and Xn ↑ X a.s. then
E [Xn |G] ↑ E [X |G]
a.s..
(e) Fatou: If 0 ≤ Xn , for all n, then
1. Stoch. pr.,
filtrations
E [lim inf Xn |G] ≤ lim inf E [Xn |G].
n
n
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
(f) Dominated convergence (DOM): If |Xn | ≤ Y , for all n ≥ 1 and
Y ∈ L1 (i.e. integrable) and lim Xn = X a.s. then
lim E [Xn |G] = E [X |G]
n
a.s.
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
(g) Jensen’s inequality: Let ϕ : R → R be a convex function such
that ϕ(X ) is integrable. Then
ϕ (E [X |G]) ≤ E [ϕ(X )|G]
1. Stoch. pr.,
filtrations
2. BM as
weak limit
(h) Tower Property: If G1 ⊆ G2 ⊆ F then
E [E [X |G2 ]|G1 ] = E [X |G1 ]
a.s.
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
(i) Taking out what is known: Let Y be G-measurable and let
E [|XY |] < ∞. Then
E [XY |G] = YE [X |G]
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
(j) Role of independence: If X is independent of G, then
E [X |G] = E [X ]
6. Martingales and the discrete time stochastic
integral
1. Stoch. pr.,
filtrations
Definition 6.1
A stochastic process (Xt )t≥0 on (Ω, F, (Ft )t≥0 , P) is a martingale if
X is adapted and E [|Xt |] < ∞, for all t ≥ 0, and, for s < t,
2. BM as
weak limit
E [Xt |Fs ] = Xs
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
a.s.
(6.1)
If the = in (6.1) is replaced by ≤ (≥) the process is called
supermartingale (submartingale).
Remark:
7. Discrete
stoch. integral
1
(6.1) is equivalent to E [Xt − Xs |Fs ] = 0 a.s.
8. Refl. princ.,
pass.times
2
Suppose (Xt )∞
n=0 is a stochastic process in discrete time adapted
to (Fn )∞
then
the martingale property reads as: for all n ≥ 1
n=0
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
E [Xn |Fn−1 ] = Xn−1
a.s.
Example 6.2 (Random walk: sum of independent r.v.)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
Let εk , k ≥ 1, be independent with E [εk ] = 0. Let
Fn = σ{ε1 , . . . , εn }, F0 = {∅, Ω}. Define X0 = 0 and, for n ≥ 1,
Xn =
n
X
εk .
k=1
Then (Xn )n≥0 is a martingale for (Fn )n≥0 .
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Example 6.3 (Product of independent r.v.)
Let εk , k ≥ 1, be non-negative and independent with E [εk ] = 1. Let
Fn = σ{ε1 , . . . , εn }, F0 = {∅, Ω}. Define X0 = 1 and, for n ≥ 1,
Xn =
n
Y
εk .
k=1
Then (Xn )n≥0 is a martingale for (Fn )n≥0 .
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Example 6.4
3. Gaussian p.
Let X be a r.v. on (Ω, F, P) such that E [|X |] < ∞. Let Ft )t≥0 be
any filtration with Ft ⊆ F, for each t ≥ 0. Define
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Xt = E [X |Ft ].
Then (Xt )t≥0 is a martingale for (Ft )t≥0 .
7. Stochastic integral in discrete time
Given is a discrete filtered probability space (Ω, F, (Fn )n≥0 , P).
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 7.1
A discrete stochastic process (Hn )∞
n=1 is called predictable if
Hn : (Ω, Fn−1 ) → (R, B(R))
is measurable. (Short: Hn is Fn−1 -measurable.)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 7.2
∞
Let (Xn )∞
n=0 be an adapted stochastic process and (Hn )n=1 be a
predictable stochastic process on (Ω, F, (Fn )n≥0 , P). Then the
stochastic integral in discrete time at time n ≥ 1 is given by
(H · X )n =
n
X
Hk (Xk − Xk−1 ).
k=1
We define (H · X )0 = 0. The stochastic integral process is given by
((H · X )n )∞
n=0 .
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 7.3
Let H be bounded and predictable and X be a martingale on
(Ω, F, (Fn )n≥0 , P). Then the stochastic integral process
((H · X )n )∞
n=0 is a martingale.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Theorem 7.4 (Doob’s Optional Stopping Theorem)
Let (Mt )t≥0 be a martingale in continuous time on
(Ω, F, (Ft )t≥0 , P). Then, for a bounded stopping time T it holds
that
E [MT ] = M0 .
More generally, if S ≤ T are bounded stopping times, then
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
E [MT |FS ] = MS
a.s.
8. Reflection principle, passage times,
running maximum/minimum
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Definition 8.1 (Markov property)
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Brownian motion satisfies the Markov property, that means, for all
t ≥ 0 and s > 0:
P(Wt+s ≤ y |Ft ) = P(Wt+s ≤ y |Wt )
a.s.
Reflection principle: Recall that Ta = inf{t > 0 : Wt = a}.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
P(Ta < t) =
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
P(Ta < t) = 2P(Wt > a)
(8.1)
1. Stoch. pr.,
filtrations
In all the following (Wt )t≥0 is a standard Brownian motion on
(Ω, F, (Ft )t≥0 , P).
2. BM as
weak limit
Theorem 8.2 (Strong Markov property)
3. Gaussian p.
The BM (Wt )t≥0 satisfies the strong Markov property: for each
stopping time τ with P(τ < ∞) = 1 it holds that
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
Ŵt = Wτ +t − Wτ ,
,t ≥ 0
is a standard Brownian motion independent of Fτ .
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
We will see later that P(Ta < ∞) = 1 for all a ∈ R.
Theorem 8.3 (Reflection principle)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Let a 6= 0 and Ta be the passage time for a.
(
Wt
for t ≤ Ta
W̃t =
2WTa − Wt = 2a − Wt for t ≥ Ta
Then (W̃t )t≥0 is again a standard Brownian motion.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Lemma 8.4
P(supt≥0 Wt = +∞ and inf t≥0 Wt = −∞) = 1
Theorem 8.5 (Recurrence)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
P(Ta < ∞) = 1 for all a ∈ R.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 8.6 (Distribution of passage time)
Let a 6= 0. The density of Ta is given by
3
a2
|a|
fTa (t) = √ t − 2 e − 2t ,
2π
which is the density of an invers-gamma-distribution with parameters
1
a2
2 and 2 . In particular, we have that E [Ta ] = +∞.
Running maximum and minimum of Brownian motion:
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Let Mt = max0≤s≤t Ws and mt = min0≤s≤t Ws .
Theorem 8.7
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1
For each a > 0:
P(Mt ≥ a) = P(Ta ≤ t) = P(Ta < t) = 2P(Wt > a).
2
For each a < 0: P(mt ≤ a) = 2P(Wt < a).
Theorem 8.8 (Joint distribution)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Let y ≥ x ≥ 0. Then
P(Wt ≤ x, Mt ≥ y ) = P(Wt ≥ 2y − x).
This implies that the joint distribution of (Wt , Mt ) has the following
density: for y ≥ 0, x ≤ y
r
2 (2y − x) −(2y −x)2
e 2t .
fW ,M (x, y ) =
3
π
t2
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Let a < 0 < b and let τ be the first time to leave an interval (a, b)
(exit time), i.e.,
τ = inf{t > 0 : Wt ∈
/ (a, b)}.
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
We have that τ = min(Ta , Tb ) = Ta ∧ Tb .
Theorem 8.9 (Exit time)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Let a < 0 < b and τ = Ta ∧ Tb as above. Then P(τ < ∞) = 1 and
E [τ ] = |ab|. Moreover
−a
|a|
=
|a| + b
−a + b
b
b
P(Wτ = a) =
=
|a| + b
−a + b
P(Wτ = b) =
9. Quadratic variation and path properties of BM
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Definition 9.1
A stochastic process has finite quadratic variation if there exists an
a.s. finite stochastic process ([X , X ]t )t≥0 such that for all t and
n
n
partitions πn = {t0n , . . . , tm
} with 0 = t0n < t1n < · · · < tm
= t and
n
n
n
n
δn = maxi=1,...,mn (ti − ti−1 ) → 0 we have that, for n → ∞,
Vt2 (πn ) :=
mn
X
i=1
n
(Xtin − Xti−1
)2 → [X , X ]t
in probability.
(9.1)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 9.2
Let (Wt )t≥0 be a Brownian motion. Then Vt2 (πn ) → t in L2 , that
means
lim E [(Vt2 (πn ) − t)2 ] = 0.
n→∞
Therefore it follows that Vt2 (πn ) → t in probability. Hence
[W , W ]t = t a.s.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Corollary 9.3
(Wt )t≥0 has a.s. paths of infinite variation on each interval.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Corollary 9.4
For almost all ω the path t 7→ Wt (ω) is not monotone in each
interval.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Remark 9.5
Almost all paths of (Wt )t≥0 are not differentiable on each interval.
10. Stochastic integral by Ito
1. Stoch. pr.,
filtrations
For the whole chapter let (Wt )t≥0 be a standard Brownian motion on
(Ω, F, (Ft )t≥0 , P). Everything will be based on this filtered
probability space.
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Recall:
Riemann integral:
Z b
f (t)dt = lim
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
a
t0n
δn →0
n
X
n
f (ξin )(tin − ti−1
)
i=1
n
where a = < t1n < · · · < tnn = b, ti−1
≤ ξin ≤ tin and
n
n
δn = maxi=1,...,n (ti − ti−1 ).
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Riemann-Stieltjes-integral: Let f be a continuous function and g be
a function of bounded variation. Then the following limit exists and
is called R-S-integral of f with respect to g :
Z
b
f (t)dg (t) := lim
a
δn →0
n
X
n
n
f (ti−1
)(g (tin ) − g (ti−1
)),
i=1
n
where a = t0n < t1n < · · · < tnn = b and δn = maxi=1,...,n (tin − ti−1
).
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Theorem 10.1
Fix [0, t] and let, for each n, 0 = t0n < t1n < · · · < tnn = t. If g is a
function such that
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
lim
δn →0
n
X
n
n
f (ti−1
)(g (tin ) − g (ti−1
))
i=1
exists for each continuous function f : [0, t] → R, then g is of
bounded variation on [0, t].
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Remark 10.2
No pathwise R-S-integral for Brownian motion!!!
Ito integral with simple integrands:
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Definition 10.3
A stochastic process (Xt )0≤t≤T is called simple predictable
integrand if
3. Gaussian p.
Xt = ξ0 1I[t0 ,t1 ] (t) +
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
n−1
X
ξi 1I(ti ,ti+1 ] (t),
i=1
where 0 = t0 < t1 < · · · < tn = T and ξ0 , . . . , ξn−1 are random
variables such that ξi is Fti -measurable and E [ξi2 ] < ∞, for
i = 1, . . . , n − 1.
For each t ≤ T the Ito integral at time t for a simple integrand X is
given by:
Z t
n−1
X
Xs dWs =: (X · W )t =
ξi (Wti+1 ∧t − Wti ∧t ).
0
i=0
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Remark 10.4
If t < T is such that tk ≤ t < tk+1 ≤ tn = T , this means
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
(X · W )t =
k−1
X
ξi (Wti+1 − Wti ) + ξk (Wt − Wtk ).
i=0
For T this means
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
(X · W )T =
n−1
X
i=0
ξi (Wti+1 − Wti )
Theorem 10.5 (Properties of Ito integral for simple integrands)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
1
Z
3. Gaussian p.
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
T
Z
(αXt + βYt ) dWt = α
0
4. Stopping
times
5. Cond.
expectation
Linearity: X , Y simple, α, β constants, then
T
Z
Xt dWt + β
0
T
Yt dWt .
0
RT
Let 0 ≤ a < b. Then 0 1I(a,b] (t)dWt = Wb − Wa and
RT
Rb
1I(a,b] (t)Xt dWt = a Xt dWt .
0
RT
3 Zero mean: E [ 0 Xt dWt ] = 0
2 R
RT
T
4 Isometry: E
Xt dWt
= 0 E [Xt2 ]dt
0
2
(More) general integrands:
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
Lemma 10.6
Let (Xt )0≤t≤T be a bounded and progressively measurable stochastic
process. Then there exists a sequence of simple predictable processes
(Xtm )0≤t≤T , m ≥ 1, such that
"Z
#
T
m→∞
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
|Xtm − Xt |2 dt = 0.
lim E
0
(10.1)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Lemma 10.7
Let (Xt )0≤t≤T be a progressively measurable stochastic process such
that
"Z
#
T
2
E
Xt dt < ∞.
(10.2)
0
Then there exists a sequence of simple predictable processes
(Xtm )0≤t≤T , m ≥ 1, such that (10.1) holds, i.e.,
"Z
#
T
|Xtm − Xt |2 dt = 0.
lim E
m→∞
0
Theorem 10.8 (Ito integral for general integrands with (10.2))
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
Let (Xt )0≤t≤T be a progressively measurable stochastic process such
RT
that (10.2) holds, i.e., E [ 0 Xt2 dt] < ∞. Then the stochastic
RT
integral 0 Xt dWt is defined and has the following properties:
1
4. Stopping
times
Z
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
T
Z
(αXt + βYt ) dWt = α
5. Cond.
expectation
6. Martingales
Linearity: X , Y as above, then
0
T
Z
Xt dWt + β
0
T
Yt dWt .
0
RT
Let 0 ≤ a < b. Then 0 1I(a,b] (t)dWt = Wb − Wa and
RT
Rb
1I(a,b] (t)Xt dWt = a Xt dWt .
0
RT
3 Zero mean: E [ 0 Xt dWt ] = 0
2 R
RT
T
4 Isometry: E
Xt dWt
= 0 E [Xt2 ]dt
0
2
Theorem 10.9 (Ito integral for even more general integrands)
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
Let (Xt )0≤t≤T be a progressively measurable stochastic process such
that
!
Z T
2
P
Xt dt < ∞ = 1.
(10.3)
0
RT
Then the stochastic integral 0 Xt dWt is defined and has satisfies
(1) and (2) of Theorem 10.8.
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Remark 10.10
Attention: in this case (3) Zero-Mean and (4) Isometry are not
satisfied in general!!! If, additionally the stronger condition (10.2)
holds, then we are in the case of Theorem 10.8 and Zero-Mean and
Isometry hold.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
Corollary 10.11
RT
Let X be continuous and adapted. Then 0 Xt dWt exists. In
particular, for any continuous function f : R → R the stochastic
RT
integral 0 f (Wt )dWt exists.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Remark 10.12
The Ito integral is not monotone.
Ito integral process:
1. Stoch. pr.,
filtrations
Let X be progressively measurable such that (10.3) holds, i.e.,
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Z
P(
T
Xs2 ds < ∞) = 1.
0
Rt
Then 0 Xs dWs is defined for all t ≤ T . This gives a stochastic
process (It )0≤t≤T with
Z
t
Xs dWs .
It =
0
There exists a modification with continuous paths: we always take
this modification.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Definition 10.13
A martingale (Mt )0≤t≤T is called quadratic integrable on [0, T ] if
sup E [Mt2 ] < ∞.
t∈[0,T ]
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 10.14
Let X be progressively measurable such that (10.2) holds, i.e.,
RT
Rt
E [ 0 Xs2 ds] < ∞. Then (It )0≤t≤T , where It = 0 Xs dWs , is a
quadratic integrable martingale.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Corollary 10.15
Let f : R → R be a continuous
and bounded function. Then
Rt
(It )0≤t≤T , where It = 0 f(Ws )dWs , is a quadratic integrable
martingale.
Quadratic variation and covariation of Ito integrals:
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Theorem 10.16
Let X be progressively measurable such that (10.3 holds. The
Rt
quadratic variation of 0 Xs dWs satisfies
Z
t
Z
Xs dWs ,
0
for all 0 ≤ t ≤ T .
0
t
Z t
Xs dWs =
Xs2 ds
0
a.s.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
Rt
Rt
Covariation: Let It = 0 Xs dWs and Ĩt = 0 X̃s dWs , for 0 ≤ t ≤ T ,
and progressively measurable processes X , X̃ such that (10.3) holds.
The covariation of I and Ĩ is given by
4. Stopping
times
5. Cond.
expectation
[I, Ĩ]t :=
1
[I + Ĩ, I + Ĩ]t − [I, I]t − [Ĩ, Ĩ]t .
2
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Corollary 10.17
It holds that [I, Ĩ]t =
Rt
0
Xs X̃s ds a.s.
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
Remark 10.18
Analogously to the quadratic variation the covariation of two
stochastic processes Y and X can be defined as follows:
4. Stopping
times
5. Cond.
expectation
[Y , X ]t = lim
n→∞
mn
X
n
n
(Ytin − Yti−1
)(Xtin − Xti−1
),
i=1
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
n
in probability where 0 = t0n < t1n < . . . tm
= t and
n
n
maxi=1,...,mn (tin − ti−1
) → 0 for n → ∞.
1. Stoch. pr.,
filtrations
Integration by parts:
2. BM as
weak limit
Theorem 10.19 (Integration by parts)
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
Let X , Y be continuous and adapted processes such that (10.3) holds
for both processes. Then the following holds, for each 0 ≤ t ≤ T ,
Z t
Z t
Xt Yt = X0 Y0 +
Xs dYs +
Ys dXs + [X , Y ]t .
0
0
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
Notation: d(Xt Yt ) = Xt dYt + Yt dXt + d[X , Y ]t
11. Ito’s formula: change of variables
1. Stoch. pr.,
filtrations
2. BM as
weak limit
3. Gaussian p.
4. Stopping
times
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
1. Stoch. pr.,
filtrations
2. BM as
weak limit
Theorem 11.1
Let (Wt )t≥0 be a Brownian motion and let f : R → R be a twice
continuously differentiable function (f ∈ C 2 (R)). Then, for each t,
Z
3. Gaussian p.
4. Stopping
times
f (Wt ) − f (W0 ) =
0
t
f 0 (Ws )dWs +
1
2
Z
t
f 00 (Ws )ds.
0
5. Cond.
expectation
6. Martingales
7. Discrete
stoch. integral
8. Refl. princ.,
pass.times
9. Quad.var.,
path prop.
10. Ito integral
11. Ito’s
formula
In differential notation (11.1) reads as follows:
1
df (Wt ) = f 0 (Wt )dWt + f 00 (Wt )dt
2
(11.1)