Stochastische Prozesse

Stochastische Prozesse

Vorlesung: Prof. Dr. Thorsten Schmidt Exercise: Tolulope Fadina

http://www.stochastik.uni-freiburg.de/lehre/2015WiSe/inhalte/2015SoSeStochastische prozesse

Exercise 1

Submission: 22-10-2015

Let (Ω,F,P) be a probability space and (E,E) be a measurable space. LetX = (X_n)n≥0 be a sequence of random variables taking value in E. We callX a stochastic process in E.

A ltration (F_n)_n is an increasing family of sub σ−algebras of F. i.e., F_n ⊆ F_n+1 for all n.

We can think of F_n as the information available to us at time n. Every process has a natural ltration (F_n^X)n given byF_n^X =σ(X_k, k≤n).

The processX is called adapted to the ltration(F_n)_n, ifX_n isF_n-measurable for all n. Every process is adapted to a natural ltration. We sayX is integrable ifXn is integrable for alln.

Denition 1 (Martingale). A sequenceξ1, ξ2,· · · ,of random variables is called a martingale with respect to the ltration F₁,F₂,· · · ,if

(1) ξ_n is integrable for each n= 1,2,· · · (11) ξ1, ξ2,· · · ,is adapted toF₁,F₂,· · ·, (111) E[ξ_n+1|F_n] =ξ_n a.s.for each n= 1,2,· · ·

Denition 2. Let(ξ_k,k≥1)be i.i.d. (independent and identically distributed) random variables.

Then

S_n=

n

X

k=1

ξ_k, n∈N,

is a random walk. Random walks have stationary and independent increments, ξ_k=S_k−Sk−1 k≥1.

Stationary simply implies that the (ξ_k)k≥1 have identical distribution.

Denition 3. A process Xn, n ∈ N with stationary independent increments is called a Lévy process. i.e., the increment X_nk −X_nk−1 are independent and X_nk −X_nk−1 ∼ X_nk−nk−1, for k= 1,· · ·, n

Denition 4 (Markov chain). A discrete process {X_n, n = 0,1,· · · } with discrete state space Xn∈ {0,1,2,· · · }is a Markov chain if it has the Markov property

P[Xn+1 =j|X_n=i, Xn−1 =in−1,· · ·, X0 =i0] =P[Xn+1=j|X_n=i].

Problem 1. Let ξ ∈L¹(Ω,F,P) and H ⊂ G beσ−algebras. Show that

E[E[ξ|G]|H] =E[ξ|H] a.s. (1)

Problem 2. Show that if ξ = (ξ_n)n≥1 is a martingale with respect toF = (F_n)n≥1, then E(ξ1) =E(ξ2) =· · ·.

Hint: What is the expectation of E(ξn+1|F_n)?

Problem 3. Suppose thatξ= (ξ_n)n≥1is a martingale with respect to the ltrationG= (G_n)n≥1. Show thatξ is a martingale with respect to the ltration

H_n=σ(ξ1,· · ·,· · · , ξn)

Hint: Observe thatH_n⊂ G_n and use the tower property of conditional expectation, (1).

Problem 4. Let ξ = (ξ_k)k≥1 be independent and inL¹ (see Denition (2)) show that

S_n⁰ =

n

X

k=1

(ξ_k−E[ξ_k])

satises the Martingale property.

Problem 5. Given a martingale (Sn)n≥1, show that

E[S_n|F_m] =E[S_n|S_m], form < n which implies the Markov property.