Stochastische Differentialgleichungen

Mathematisch-

Naturwissenschaftliche Fakultät

Fachbereich Mathematik

Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee

Stochastische Differentialgleichungen

Summer Semester 2017 Tübingen, 02.06.2017

Homework 7

Problem 1. Let X =

Xt; 0 ≤ t ≤ T be the Itô integral Xt := Rt

0fsdWs. Compute the quadratic variation process

X

=

X

t; 0≤t≤T . Problem 2.Let0≤t <1. Consider the SDE

dX_t=− X_t

1−tdt+dW_t, X₀= 0.

i) Show that its solution is

Xt= (1−t) Z t

0

1

1−sdWs. It is known asBrownian bridge.

ii) Deduce that

Xt: 0≤t <1 is a Gaussian process. Compute its mean and covariance function.

iii) Show that 0 =L²−lim

t↑1 X_t.

Problem 3.Letb :Rⁿ → Rⁿ andσσσ :Rⁿ → R^n×n be locally Lipschitz functions, i.e., for everyN ≥0 there existsC_N >0such that

kb(x)−b(y)k_Rⁿ+kσσσ(x)−σσσ(y)k_L(Rn,Rⁿ)≤CNkx−yk_Rⁿ ∀x,y∈[−N, N]ⁿ.

i) Show that for everyx∈Rⁿ, we can find a stopping timeτx, almost surely positive, and a stocha- stic process

X^x_t; 0≤t < τx such that

X^x_t =x+ Z t

0

b(X^x_s)ds+ Z t

0

σ

σσ(X^x_s)dW_s ∀t < τ_x. (1)

ii) Show that the process

X^x_t; 0 ≤ t < τ_x is unique in the sense that if τ˜_x is an almost surely positive stopping time and if

Y^x_t; 0≤t <τ˜x is a stochastic process such that

Y_t^x=x+ Z t

0

b(Y_s^x)ds+ Z t

0

σσσ(Y^x_s)dW_s ∀t <τ˜_x,

thenτ˜x ≤τx, andP-a.s.,

Y_t^x1{t<τ˜x} =X^x_t1{t<˜τx} ∀t≥0.

The process

X^x_t; 0≤t < τ_x is called the solution of the SDE (1) up to the explosion timeτ_x.

Problem 4. (Part I)LetI_k =

t_j ^J_j=0 be an equi-distant mesh of sizek = k^J > 0. In the lecture, we

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use the Euler method

Y^j+1−Y^j =b(Y^j)k+σσσ(Y^j)∆jW (0≤j≤J−1)

to construct an Rⁿ-valued solution of the martingale problem MP(σσσσσσ^>,b, µ) on [0, T], where both b andσσσare continuous on[0, T]×Rⁿ, with asymptotic linear growth, i.e.,

kb(x)k_Rn +kσσσ(x)k_L(Rm,Rⁿ)≤C_T 1 +kxk_Rn

∀x∈Rⁿ.

Letn=m= 1.

i) Prove the existence of a constantC >0such that

sup

k>0

0≤j≤Jmax E h

kY^jk⁴

Rⁿ

i

≤C

1 +E h

kY⁰k⁴

Rⁿ

i

exp(CT).

ii) For everyk= ^T_J, letY^(k)∈L²

F(Ω;C([0, T];Rⁿ))denote the ‘ continuified Euler iterate’. Show that sup

k>0

E h

sup

0≤s≤T

kY_s^(k)k⁴

Rⁿ

i

≤C.

Date of submission: 14.06.2017.

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