Stochastische Differentialgleichungen

Mathematisch-

Naturwissenschaftliche Fakultät

Fachbereich Mathematik

Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee

Stochastische Differentialgleichungen

Summer Semester 2017 Tübingen, 22.06.2017

Homework 9

Problem 1.Let{Yⁿ :n≥1}be a sequence of stochastic processes such that for each fixedt,Y_tⁿ is uniformly integrable and for eachn,Y_tⁿis a martingale with respect to its own filtration. IfYⁿconverges toY in probability asngoes to∞, then show thatY is a martingale with respect to its own filtration.

Hint:It suffices to prove that for any0≤s1 < s2,· · ·< sm < s < t≤T, the following equation

E h

Yt m

Y

i=1

fi(Ysi) i

=E h

Ys m

Y

i=1

fi(Ysi) i

holds for allf_i(·)∈C_b(R), i= 1,2,· · ·, m, m≥1.

Problem 2.LetX=

X_t;t≥0 be anRⁿ-valued Markov process on(Ω,F,F,P). We defined transition probabilities

Ps,t[x,A] :=

S_s,tχ_A

(x) =P h

X_t∈A

X_s=xi

∀x∈Rⁿ ∀A ∈ B(Rⁿ) ∀0≤s≤t <∞.

a) Show thatPs,t[x,·]is a probability measure on Rⁿ,B(Rⁿ) . b) Show that

S_s,tf(x) = Z

Rⁿ

f(y)Ps,t[x, dy] ∀f ∈ B_b(Rⁿ) ∀x∈Rⁿ.

Problem 3. Let W be an R-valued Wiener process on (Ω,F,P). Show that W is a Markov process with semigroup

S₀^W =1, S_t^Wf(x) = Z

R

f(y)e^{−(x−y)22t}

√

2πt dy ∀t >0 ∀x∈R.

Date of submission: 28.06.2017.

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