Stochastische Differentialgleichungen

Mathematisch-

Naturwissenschaftliche Fakultät

Fachbereich Mathematik

Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee

Stochastische Differentialgleichungen

Summer Semester 2017 Tübingen, 26.04.2017

Homework 2

Problem 1.LetT >0, and(Ω,F,P)be a probability space. In the lecture, we introduced the Lagrange interpolation

Ye_k,h(t); 0≤t≤T via

Ye_k,h(t) = tj+1−t

k Y_k,h(tj) +t−tj

k Y_k,h(tj+1) ∀tj ≤t < tj+1, whereY_k,h(tj) =Pj

l=1η_lis the random walk, andtj =j^T_J =:jkare the mesh points of the equi-distant meshGk :=

t_j ^J_j=0 covering[0, T]. Letf ∈C_b(R). Show that the iterates

u^{j J}_j=0 ⊂ C_b(R)be given by

u^j(x) =E h

f x+Ye_k,^√_k(t_j)i

∀x∈R solves

(_uj(x)−u^j−1(x)

k = ^uj−1(x+

√k)−2u^j−1(x)+u^j−1(x−√ k)

2k (1≤j≤J)

u⁰(x) =f(x) ∀x∈R, (1)

where (1) is a finite difference discretization of the linear heat equation.

Problem 2.Lett >0. In the lecture, we define the quadratic variation of a Wiener processW Q^W(t) =L²−lim

J↑∞Q^W_kJ(t), where Q^W_kJ(t) = PJ

j=1

W(t^J_j)− W(t^J_j−1)

2 on the equi-distant mesh of size k > 0 covering [0, t]

I_kJ =

0, t^J₁,· · · , t^J_J wheret^J_j = ^jt_J =:jk. Let

I_kl;l∈N be a sequence of equi-distant meshes such thatP∞

l=1k^l<∞. Show that

Q^W_kl(t)→t P-a.s. (l↑ ∞)

Hint:Use Tschebycheff’s inequality and Borel-Cantelli lemma to transfer the corresponding L²-convergence result from the lecture.

Problem 3. Let0 = t^J₀ < t^J₁ < · · · < t^J_J = T, wheret^J_j = ^jT_J be an equi-distant mesh of sizek = ^T_J covering[0, T]. Find the following limits

a). L²−lim

J↑∞

J−1

X

j=0

W(t^J_j)h

W(t^J_j+1)−W(t^J_j)i ,

b). L²− lim

J↑∞

J−1

X

j=0

W(t^J_j+1)h

W(t^J_j+1)−W(t^J_j)i .

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Problem 4.Letf ∈M_step² such thatf(t) =PJ−1

j=0 η_j1[tj,tj+1)(t). Consider the stochastic integral

I(f) =

J−1

X

j=1

η_j

W(t_j+1)−W(t_j) .

Show that

E h

I(f)

2i

=E hZ ∞

0

|f(s)|²dsi .

Date of submission: 03.05.2017.

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