Exercises, 7th Mai

Stochastic Processes II Summer term 2008 (Stochastische Analysis)

Prof. Dr. Uwe K¨uchler Dr. Irina Penner

Exercises, 7th Mai

4.1 (3 points) We denote by h _n (x, t), n = 0, 1, . . . the Hermite-polynomials defined by

exp(αx − 1

2 α ² t) = X ∞

n=0

α ⁿ

n! h _n (x, t), x ∈ R , t ≥ 0, i.e.

h _n (x, t) = d ⁿ

dα ⁿ exp(αx − 1

2 α ² t) | _α=0 .

Show that for a continuous function (x _t ) _t≥0 with continuous quadratic variation hxi and x ₀ = 0 it holds

h _n (x _t , hxi _t ) = n Z _t

0

h _n−1 (x _s , hxi _s )dx _s

= n!

Z _t

0

. . . Z _t

0

dx t

. . . dx t

, n = 1, 2, . . . . In particular the functions H _n := h _n (x, hxi), n = 0, 1, . . . solve the Itˆo- equation

dH _n = nH _n−1 dx.

4.2 (4 points) Let (X _t , t ≥ 0) be a geometric Brownian motion, i.e.

X _t = x ₀ exp(σB _t + αt), t ≥ 0

where x ₀ > 0 is a real number, (B _t , t ≥ 0) a Standard Brownian motion and σ > 0, α ∈ R ₁ .

i) Calculate all moments E[X _t ^p ], t ≥ 0, p ∈ R.

ii) For which σ, α the process X _t , t ≥ 0) is a martingale?

iii) Prove that for every n ≥ 1 the process (h _n (B _t , t), t ≥ 0) is a martingale. (Notation as in exercise 4.1)

4.3 (3 points) Let (B _t , t ≥ 0) be a Standard Brownian motion.

i) Show that for α, σ, x ₀ ∈ R the solution of

dX _t = −aX _t dt + σdB _t , X ₀ = x ₀ is given by

X t = e ^−at (x 0 + Z t

0

e ^as σdB s )

= e ^−at x 0 + σB t − aσ Z t

0

e ^−a(t−s) B s ds, t ≥ 0.

ii) Calculate E[X _t ], V ar[X _t ], Cov[X _s , X _t ] for s, t ≥ 0.

((X _t ) is called the Ornstein-Uhlenbeck-process.)

4.4 (3 points) Consider the Brownian Bridge

Z _t := (1 − t) Z t

0

1

1 − s dB _s , t ∈ [0, 1], Z ₁ := 0 (with (B _t ) _0≤t≤1 standard BM).

Show that Z solves the Itˆo-equation

dZ _t = dB _t − Z _t

1 − t dt

on (0, 1) with Z ₀ = Z ₁ = 0.