Stochastische Differentialgleichungen

Mathematisch-

Naturwissenschaftliche Fakultät

Fachbereich Mathematik

Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee

Stochastische Differentialgleichungen

Summer Semester 2017 Tübingen, 24.05.2017

Homework 6

Problem 1.LetT >0, andW be anRⁿ-valued Wiener process on a given filtered probability space (Ω,F,F,P). SupposeAAA∈Lspd(Rⁿ,Rⁿ), andσσσ ∈L(Rⁿ,Rⁿ). Prove that

(dX_t= A

AAX_t− |X_t|²X_t+X_t

dt+σσσdW_t (t∈(0, T]) X₀=~x∈Rⁿ

has a strong solution on[0, T].

Problem 2.In the lecture, we constructed a strong solutionX≡

X_t; 0≤t≤T of the SDE (dX_t=b(X_t) dt+σσσ(X_t)dW_t (t∈(0, T])

X₀=~x∈Rⁿ. (1)

a) Let(b, σσσ)be Lipschitz, andp∈N. Show the existence of a constantC_p ≡C_p(T)>0such that

sup

0≤t≤TE h

kX_tk^p

Rⁿ

i

≤Cp.

b) Verify the same result for(b, σσσ)such that Assumption1is valid.

Problem 3.In case that(b, σσσ)satisfy Assumption1, the Banach space C_F([0, T];L²(Ω))was used in the lecture to construct the strong solution of the (truncated) SDE via Banach fixed point theorem. Let (b, σσσ)be Lipschitz. Is it possible to construct a strong solution of (1) already inL^p

F Ω;C([0, T];Rⁿ) , p≥ 2accordingly via the iteration(n∈N0)

(dXⁿ_t =b(Xⁿ_t) dt+σσσ(Xⁿ⁻¹_t )dW_t (t∈(0, T]) Xⁿ₀ =~x∈Rⁿ

whereX⁰ ≡~x?

Date of submission: 31.05.2017.

Seite 1/1