1. Stochastic Processes and ltrations · 4. Stopping times 5. Cond. expectation 6. Martingales 7....

1. Stoch. pr.,filtrations

2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales

7. Discretestoch. integral

8. Refl. princ.,pass.times

9. Quad.var.,path prop.

10. Ito integral

11. Ito’sformula

1. Stochastic Processes and filtrations

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.

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).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 1.1

Random walk:Let εn, n ≥ 1, be a sequence of iid random variables with

εn =

1 with probability 1

2

−1 with probability 12

Define X0 = 0 and, for k = 1, 2, . . . ,

Xk =k∑

i=1

εi


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

For a fixed ω ∈ Ω the function

t 7→ Xt(ω)

is called (sample) path (or realization) associated to ω of the

stochastic process X .


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Let X and Y be two stochastic processes on (Ω,F).

Definition 1.2

(i) X and Y have the same finite-dimensional distributions if, forall 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,∞),

P(Xt = Yt) = 1.

(iii) X and Y are indistinguishable, if

P(Xt = Yt for every t ∈ [0,∞)) = 1.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 1.3


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 1.4

A stochastic process X is called continuous (right continuous), ifalmost all paths are continuous (right continuous), i.e.,

P(ω : t 7→ Xt(ω) continuous (right continuous)) = 1

.

Proposition 1.5

If X and Y are modifications of each other and if both processes area.s. right continuous then X and Y are indistinguishable.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Let (Ω,F) be given and Ft , t ≥ 0 be an increasing family ofσ-algebras (i.e., for s < t, Fs ⊆ Ft), such that Ft ⊆ F for all t.That’s a

filtration.

For example:

Definition 1.6

The filtration FXt generated by a stochastic process X is given by

FXt = σXs , s ≤ t.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 1.7

A stochastic process is adapted to a filtration (Ft)t≥0 if, for eacht ≥ 0

Xt : (Ω,Ft)→ (R,B(R))

is measurable. (Short: Xt is Ft-measurable.)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 1.8 (”The usual conditions”)

Let Ft+ = ∩ε>0Ft+ε. We say that a filtration (Ft)t≥0 satisfies theusual 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 ofP-nullsets.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

2. Brownian motion as a weak limit of randomwalks

Example 2.1

Scaled random walk: for tk = kN , k = 0, 1, 2, . . . , we set

XNtk :=

1√N

Xk =1√N

k∑i=1

εi


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 2.2 (Brownian motion)

A standard, one-dimensional Brownian motion is a continuousadapted process W on (Ω,F , (Ft)t≥0,P) with W0 = 0 a.s. suchthat, for all 0 ≤ s < t,

1 Wt −Ws is independent of Fs

2 Wt −Ws is normally distributed with mean 0 and variance t − s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Recall the scaled random walk: for tk = kN , k = 0, 1, 2, . . . , we set

XNtk :=

1√N

Xk =1√N

k∑i=1

εi

By interpolation we define a continuous process on [0,∞):

XNt =

1√N

[Nt]∑i=1

εi + (Nt − [Nt])ε[Nt]+1

(2.1)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 2.3 (Donsker)

Let (Ω,F ,P) be a probability space with a sequence of iid randomvariables with mean 0 and variance 1. Define (XN

t )t≥0 as in (2.1).Then (XN

t )t≥0 converges to (Wt)t≥0 weakly.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Clarification of the statement:

1 Ω = C [0,∞)

2 On Ω = 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 (XNt )t≥0 defines a measure probability PN on Ω = C [0,∞).

Indeed,XN : (Ω,F)→ (C [0,∞),B(C [0,∞))

measurable, that means we consider the random variableω 7→ XN(ω, ·), which maps ω to a continous function in t (i.e.the path). Let A ∈ B(C [0,∞)), then

PN(A) := P(ω : XN(ω, · ∈ A).

4 So, Donsker’s theorem says that on(Ω, F) = (C [0,∞),B(C [0,∞))) we have that PN w−→ P∗.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Steps of the proof:

1 Show that the finite-dimensional distributions of XN convergeto the finite-dimensional distributions of W . That means, showthat, for all n ≥ 1, and all s1, . . . , sn ∈ [0,∞)

(XNs1,XN

s2, . . . ,XN

sn )w−→ (Ws1 ,Ws2 , . . . ,Wsn).

2 Show that (PN)N≥1 is a tight familiy of probability measures on

(Ω, F).

3 (1) and (2) imply PN w−→ P∗.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Easy partial result of step (1) of the proof:

By (CLT), for fixed t ∈ [0,∞), for N →∞,

XNt

w−→Wt

Theorem 2.4 (CLT)

Let (ξn)n≥1 be iid, mean=0, variance=σ2. Then, for each x ∈ R,

limn→∞

P

(1√n

n∑i=1

ξi ≤ x

)= Φσ(x),

where Φσ(x) is the distribution function of N(0, σ2). In other words

1√n

n∑i=1

ξiw−→ Z ,

where Z ∼ N(0, σ2).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

3. Brownian motion as a Gaussian process

Definition 3.1

A stochastic process (Xt)t≥0 is called Gaussian process, if for eachfinite familiy of time points t1, . . . , tn the vector (Xt1 , . . . ,Xtn) hasa 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).

Theorem 3.2

A Brownian motion is a centered Gaussian process withγ(s, t) = s ∧ t (= min(s, t)). Conversely, a centered Gaussian processwith continuous paths and covariance function γ(s, t) = s ∧ t is aBrownian motion.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

4. Stopping times

Given is a filtered probability space (Ω,F , (Ft)t≥0,P).

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 thefunction XT on T <∞ by:

XT (ω) = XT (ω)(ω)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 4.2

A random time T is a stopping time for (Ft)t≥0 if, for all t ≥ 0,

T ≤ t = ω : T (ω) ≤ t ∈ Ft .

Proposition 4.3

1 If T (ω) = t a.s. for some constant t ≥ 0, then T is a stoppingtime.

2 Every stopping time T satisfies that, for all t,

T < t ∈ Ft . (4.1)

3 If the filtration is right continuous and a random time Tsatisfies (4.1) for all t, then T is a stopping time.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Lemma 4.4

Let S and T be stopping times for (Ft)t≥0. Then S ∧ T , S ∨ T andS + 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 eventsdetermined prior to the stopping time T is given by

FT = A ∈ F : A ∩ T ≤ t ∈ Ft , for all t ≥ 0.

Remark:

1 FT is a σ-field.

2 T is FT -measurable.

3 If T (ω) = t a.s., then FT = Ft .


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 4.6

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) = inft > 0 : Wt(ω) = a. (4.2)

Then Ta is a stopping time for (Ft)t≥0.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Remark: Let S and T be stopping times for (Ft)t≥0.

1 For A ∈ FS we have that A ∩ S ≤ T ∈ FT .

2 If, moreover, S ≤ T then FS ⊆ FT .

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.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example: If (Xt)t≥0 is adapted and right-continuous then it isprogressively measurable.

Proposition 4.8

Let X be progressively measurable on (Ω,F , (Ft)t≥0) and let T be astopping time for (Ft)t≥0. Then

1 XT defined on T <∞ is an FT -measurable random variableand

2 the stopped process (XT∧t)t≥0 is progressively measurable.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

5. Conditional expected value

Definition 5.1

Let (Ω,F ,P) be a probability space and X : Ω→ R be anF − B(R)-measurable random variable such thatE [|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:E [Y 1IA] = E [X 1IA].

The G-measurable random variable Y is called conditional expectedvalue of X given G. We write Y = E [X |G] a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Remark:

1 E [X |G] is unique with respect to equality a.s.

2 Suppose that, additionally, E [X 2] <∞. Then E [X |G] is theorthogonal projection of X ∈ L2(Ω,F) onto the subspaceL2(Ω,G).

3 If G = ∅,Ω then E [X |G] = E [X ] a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 5.2

Let Z be a function on Ω. Then E [X |Z ] := E [X |σ(Z )].

Example 5.3


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 5.4

X : (Ω,F)→ (R,B(R)). Suppose Ω =⋃∞

k=1 Ωk such thatΩi ∩ Ωj = ∅ for i 6= j . Let G := σΩk , k ≥ 1.

E [X |G] =


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 5.5

Suppose (X ,Z ) has a bivariate density f (x , z).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 5.6 (A list of properties of conditional expectation)

In the following let X , Y , Xn, n ≥ 1, be integrable random variableson (Ω,F ,P). The following holds

(a) If X is G-measurable, then E [X |G] = X a.s.

(b) Linearity: E [aX + bY |G] = aE [X |G] + bE [Y |G] a.s.

(c) Positivity: If X ≥ 0 a.s. then E [X |G] ≥ 0 a.s.

(d) Monotone convergence (MON): If 0 ≤ Xn and Xn ↑ X a.s. then

E [Xn|G] ↑ E [X |G] a.s..


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

(e) Fatou: If 0 ≤ Xn, for all n, then

E [lim infn

Xn|G] ≤ lim infn

E [Xn|G].

(f) Dominated convergence (DOM): If |Xn| ≤ Y , for all n ≥ 1 andY ∈ L1 (i.e. integrable) and lim Xn = X a.s. then

limn

E [Xn|G] = E [X |G] a.s.

(g) Jensen’s inequality: Let ϕ : R→ R be a convex function suchthat ϕ(X ) is integrable. Then

ϕ (E [X |G]) ≤ E [ϕ(X )|G]


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

(h) Tower Property: If G1 ⊆ G2 ⊆ F then

E [E [X |G2]|G1] = E [X |G1] a.s.

(i) Taking out what is known: Let Y be G-measurable and letE [|XY |] <∞. Then

E [XY |G] = YE [X |G]

(j) Role of independence: If X is independent of G, then

E [X |G] = E [X ]


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

6. Martingales and the discrete time stochasticintegral

Definition 6.1

A stochastic process (Xt)t≥0 on (Ω,F , (Ft)t≥0,P) is a martingale ifX is adapted and E [|Xt |] <∞, for all t ≥ 0, and, for s < t,

E [Xt |Fs ] = Xs a.s. (6.1)

If the = in (6.1) is replaced by ≤ (≥) the process is calledsupermartingale (submartingale).

Remark:

1 (6.1) is equivalent to E [Xt − Xs |Fs ] = 0 a.s.

2 Suppose (Xt)∞n=0 is a stochastic process in discrete time adapted

to (Fn)∞n=0 then the martingale property reads as: for all n ≥ 1

E [Xn|Fn−1] = Xn−1 a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 6.2 (Random walk: sum of independent r.v.)

Let εk , k ≥ 1, be independent with E [εk ] = 0. LetFn = σε1, . . . , εn, F0 = ∅,Ω. Define X0 = 0 and, for n ≥ 1,

Xn =n∑

k=1

εk .

Then (Xn)n≥0 is a martingale for (Fn)n≥0.

Example 6.3 (Product of independent r.v.)

Let εk , k ≥ 1, be non-negative and independent with E [εk ] = 1. LetFn = σε1, . . . , εn, F0 = ∅,Ω. Define X0 = 1 and, for n ≥ 1,

Xn =n∏

k=1

εk .

Then (Xn)n≥0 is a martingale for (Fn)n≥0.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Example 6.4

Let X be a r.v. on (Ω,F ,P) such that E [|X |] <∞. Let Ft)t≥0 beany filtration with Ft ⊆ F , for each t ≥ 0. Define

Xt = E [X |Ft ].

Then (Xt)t≥0 is a martingale for (Ft)t≥0.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

7. Stochastic integral in discrete time

Given is a discrete filtered probability space (Ω,F , (Fn)n≥0,P).

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.)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 7.2

Let (Xn)∞n=0 be an adapted stochastic process and (Hn)∞n=1 be apredictable stochastic process on (Ω,F , (Fn)n≥0,P). Then thestochastic integral in discrete time at time n ≥ 1 is given by

(H · X )n =n∑

k=1

Hk(Xk − Xk−1).

We define (H · X )0 = 0. The stochastic integral process is given by((H · X )n)∞n=0.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

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.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

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 holdsthat

E [MT ] = M0.

More generally, if S ≤ T are bounded stopping times, then

E [MT |FS ] = MS a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

8. Reflection principle, passage times,running maximum/minimum

Definition 8.1 (Markov property)

Brownian motion satisfies the Markov property, that means, for allt ≥ 0 and s > 0:

P(Wt+s ≤ y |Ft) = P(Wt+s ≤ y |Wt) a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Reflection principle: Recall that Ta = inft > 0 : Wt = a.

P(Ta < t) =

P(Ta < t) = 2P(Wt > a) (8.1)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

In all the following (Wt)t≥0 is a standard Brownian motion on(Ω,F , (Ft)t≥0,P).

Theorem 8.2 (Strong Markov property)

The BM (Wt)t≥0 satisfies the strong Markov property: for eachstopping time τ with P(τ <∞) = 1 it holds that

Wt = Wτ+t −Wτ , , t ≥ 0

is a standard Brownian motion independent of Fτ .

We will see later that P(Ta <∞) = 1 for all a ∈ R.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 8.3 (Reflection principle)

Let a 6= 0 and Ta be the passage time for a.

Wt =

Wt for t ≤ Ta

2WTa −Wt = 2a−Wt for t ≥ Ta

Then (Wt)t≥0 is again a standard Brownian motion.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Lemma 8.4

P(supt≥0 Wt = +∞ and inft≥0 Wt = −∞) = 1


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 8.5 (Recurrence)

P(Ta <∞) = 1 for all a ∈ R.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 8.6 (Distribution of passage time)

Let a 6= 0. The density of Ta is given by

fTa(t) =|a|√2π

t−32 e−

a2

2t ,

which is the density of an invers-gamma-distribution with parameters12 and a2

2 . In particular, we have that E [Ta] = +∞.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Running maximum and minimum of Brownian motion:

Let Mt = max0≤s≤t Ws and mt = min0≤s≤tWs .

Theorem 8.7

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).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 8.8 (Joint distribution)

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 followingdensity: for y ≥ 0, x ≤ y

fW ,M(x , y) =

√2

π

(2y − x)

t32

e−(2y−x)2

2t .


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Let a < 0 < b and let τ be the first time to leave an interval (a, b)(exit time), i.e.,

τ = inft > 0 : Wt /∈ (a, b).

We have that τ = min(Ta,Tb) = Ta ∧ Tb.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 8.9 (Exit time)

Let a < 0 < b and τ = Ta ∧ Tb as above. Then P(τ <∞) = 1 andE [τ ] = |ab|. Moreover

P(Wτ = b) =|a||a|+ b

=−a

−a + b

P(Wτ = a) =b

|a|+ b=

b

−a + b


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

9. Quadratic variation and path properties of BM

Definition 9.1

A stochastic process has finite quadratic variation if there exists ana.s. finite stochastic process ([X ,X ]t)t≥0 such that for all t andpartitions πn = tn0 , . . . , tnmn

with 0 = tn0 < tn1 < · · · < tnmn= t and

δn = maxi=1,...,mn(tni − tni−1)→ 0 we have that, for n→∞,

V 2t (πn) :=

mn∑i=1

(Xtni− Xtni−1

)2 → [X ,X ]t in probability. (9.1)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 9.2

Let (Wt)t≥0 be a Brownian motion. Then V 2t (πn)→ t in L2, that

meanslim

n→∞E [(V 2

t (πn)− t)2] = 0.

Therefore it follows that V 2t (πn)→ t in probability. Hence

[W ,W ]t = t a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Corollary 9.3

(Wt)t≥0 has a.s. paths of infinite variation on each interval.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Corollary 9.4

For almost all ω the path t 7→Wt(ω) is not monotone in eachinterval.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Remark 9.5

Almost all paths of (Wt)t≥0 are not differentiable on each interval.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

10. Stochastic integral by Ito

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 filteredprobability space.

Recall:

Riemann integral:∫ b

a

f (t)dt = limδn→0

n∑i=1

f (ξni )(tni − tni−1)

where a = tn0 < tn1 < · · · < tnn = b, tni−1 ≤ ξni ≤ tni andδn = maxi=1,...,n(tni − tni−1).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Riemann-Stieltjes-integral: Let f be a continuous function and g bea function of bounded variation. Then the following limit exists andis called R-S-integral of f with respect to g :∫ b

a

f (t)dg(t) := limδn→0

n∑i=1

f (tni−1)(g(tni )− g(tni−1)),

where a = tn0 < tn1 < · · · < tnn = b and δn = maxi=1,...,n(tni − tni−1).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 10.1

Fix [0, t] and let, for each n, 0 = tn0 < tn1 < · · · < tnn = t. If g is afunction such that

limδn→0

n∑i=1

f (tni−1)(g(tni )− g(tni−1))

exists for each continuous function f : [0, t]→ R, then g is ofbounded variation on [0, t].

Remark 10.2

No pathwise R-S-integral for Brownian motion!!!


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Ito integral with simple integrands:

Definition 10.3

A stochastic process (Xt)0≤t≤T is called simple predictableintegrand if

Xt = ξ01I[t0,t1](t) +n−1∑i=1

ξi1I(ti ,ti+1](t),

where 0 = t0 < t1 < · · · < tn = T and ξ0, . . . , ξn−1 are randomvariables such that ξi is Fti -measurable and E [ξ2

i ] <∞, fori = 1, . . . , n − 1.

For each t ≤ T the Ito integral at time t for a simple integrand X isgiven by: ∫ t

0

XsdWs =: (X ·W )t =n−1∑i=0

ξi (Wti+1∧t −Wti∧t).


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Remark 10.4

If t < T is such that tk ≤ t < tk+1 ≤ tn = T , this means

(X ·W )t =k−1∑i=0

ξi (Wti+1 −Wti ) + ξk(Wt −Wtk ).

For T this means

(X ·W )T =n−1∑i=0

ξi (Wti+1 −Wti )


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 10.5 (Properties of Ito integral for simple integrands)

1 Linearity: X , Y simple, α, β constants, then∫ T

0

(αXt + βYt) dWt = α

∫ T

0

XtdWt + β

∫ T

0

YtdWt .

2 Let 0 ≤ a < b. Then∫ T

01I(a,b](t)dWt = Wb −Wa and∫ T

01I(a,b](t)XtdWt =

∫ b

aXtdWt .

3 Zero mean: E [∫ T

0XtdWt ] = 0

4 Isometry: E

[(∫ T

0XtdWt

)2]

=∫ T

0E [X 2

t ]dt


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

(More) general integrands:

Lemma 10.6

Let (Xt)0≤t≤T be a bounded and progressively measurable stochasticprocess. Then there exists a sequence of simple predictable processes(Xm

t )0≤t≤T , m ≥ 1, such that

limm→∞

E

[∫ T

0

|Xmt − Xt |2dt

]= 0. (10.1)


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Lemma 10.7

Let (Xt)0≤t≤T be a progressively measurable stochastic process suchthat

E

[∫ T

0

X 2t dt

]<∞. (10.2)

Then there exists a sequence of simple predictable processes(Xm

t )0≤t≤T , m ≥ 1, such that (10.1) holds, i.e.,

limm→∞

E

[∫ T

0

|Xmt − Xt |2dt

]= 0.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 10.8 (Ito integral for general integrands with (10.2))

Let (Xt)0≤t≤T be a progressively measurable stochastic process such

that (10.2) holds, i.e., E [∫ T

0X 2t dt] <∞. Then the stochastic

integral∫ T

0XtdWt is defined and has the following properties:

1 Linearity: X , Y as above, then∫ T

0

(αXt + βYt) dWt = α

∫ T

0

XtdWt + β

∫ T

0

YtdWt .

2 Let 0 ≤ a < b. Then∫ T

01I(a,b](t)dWt = Wb −Wa and∫ T

01I(a,b](t)XtdWt =

∫ b

aXtdWt .

3 Zero mean: E [∫ T

0XtdWt ] = 0

4 Isometry: E

[(∫ T

0XtdWt

)2]

=∫ T

0E [X 2

t ]dt


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 10.9 (Ito integral for even more general integrands)

Let (Xt)0≤t≤T be a progressively measurable stochastic process suchthat

P

(∫ T

0

X 2t dt <∞

)= 1. (10.3)

Then the stochastic integral∫ T

0XtdWt is defined and has satisfies

(1) and (2) of Theorem 10.8.

Remark 10.10

Attention: in this case (3) Zero-Mean and (4) Isometry are notsatisfied in general!!! If, additionally the stronger condition (10.2)holds, then we are in the case of Theorem 10.8 and Zero-Mean andIsometry hold.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Corollary 10.11

Let X be continuous and adapted. Then∫ T

0XtdWt exists. In

particular, for any continuous function f : R→ R the stochastic

integral∫ T

0f (Wt)dWt exists.

Remark 10.12

The Ito integral is not monotone.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Ito integral process:

Let X be progressively measurable such that (10.3) holds, i.e.,

P(

∫ T

0

X 2s ds <∞) = 1.

Then∫ t

0XsdWs is defined for all t ≤ T . This gives a stochastic

process (It)0≤t≤T with

It =

∫ t

0

XsdWs.

There exists a modification with continuous paths: we always takethis modification.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Definition 10.13

A martingale (Mt)0≤t≤T is called quadratic integrable on [0,T ] if

supt∈[0,T ]

E [M2t ] <∞.

Theorem 10.14

Let X be progressively measurable such that (10.2) holds, i.e.,

E [∫ T

0X 2s ds] <∞. Then (It)0≤t≤T, where It =

∫ t

0XsdWs, is a

quadratic integrable martingale.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Corollary 10.15

Let f : R→ R be a continuous and bounded function. Then(It)0≤t≤T, where It =

∫ t

0f(Ws)dWs, is a quadratic integrable

martingale.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Quadratic variation and covariation of Ito integrals:

Theorem 10.16

Let X be progressively measurable such that (10.3 holds. The

quadratic variation of∫ t

0XsdWs satisfies[∫ t

0

XsdWs ,

∫ t

0

XsdWs

]=

∫ t

0

X 2s ds a.s.

for all 0 ≤ t ≤ T .


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Covariation: Let It =∫ t

0XsdWs and It =

∫ t

0XsdWs , for 0 ≤ t ≤ T ,

and progressively measurable processes X , X such that (10.3) holds.

The covariation of I and I is given by

[I, I]t :=1

2

([I + I, I + I]t − [I, I]t − [I, I]t

).

Corollary 10.17

It holds that [I, I]t =∫ t

0XsXsds a.s.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Remark 10.18

Analogously to the quadratic variation the covariation of twostochastic processes Y and X can be defined as follows:

[Y ,X ]t = limn→∞

mn∑i=1

(Ytni− Ytni−1

)(Xtni− Xtni−1

),

in probability where 0 = tn0 < tn1 < . . . tnmn= t and

maxi=1,...,mn(tni − tni−1)→ 0 for n→∞.


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Integration by parts:

Theorem 10.19 (Integration by parts)

Let X , Y be continuous and adapted processes such that (10.3) holdsfor both processes. Then the following holds, for each 0 ≤ t ≤ T ,

XtYt = X0Y0 +

∫ t

0

XsdYs +

∫ t

0

YsdXs + [X ,Y ]t .

Notation: d(XtYt) = XtdYt + YtdXt + d [X ,Y ]t


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

11. Ito’s formula: change of variables


2. BM asweak limit

3. Gaussian p.

4. Stoppingtimes

5. Cond.expectation

6. Martingales




10. Ito integral

11. Ito’sformula

Theorem 11.1

Let (Wt)t≥0 be a Brownian motion and let f : R→ R be a twicecontinuously differentiable function (f ∈ C 2(R)). Then, for each t,

f (Wt)− f (W0) =

∫ t

0

f ′(Ws)dWs +1

2

∫ t

0

f ′′(Ws)ds. (11.1)

In differential notation (11.1) reads as follows:

df (Wt) = f ′(Wt)dWt +1

2f ′′(Wt)dt

1. Stochastic Processes and ltrations · 4. Stopping times 5. Cond. expectation 6. Martingales 7....

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Transcript of 1. Stochastic Processes and ltrations · 4. Stopping times 5. Cond. expectation 6. Martingales 7....