Stochastic Mechanics 6 CFU Part II 8.6.2009 Exercise 1 Let W

Stochastic Mechanics 6 CFU Part II 8.6.2009

Exercise 1Let W^t be a Brownian motion. Illustrate Ito’s formula and use it to calculate

a d(Xt²ⁿ) for n ≥ 1, where dX^t= 3t³dWt

b (dWt²ⁿ⁺¹) for n ≥ 1 c R^T

0 Wt⁴dWt

Exercise 2 Solve the following stochastic differential equation dXt = [3 e^2Xt + 5]dt + dW^t

using the substitution y = h(x) = e^−2x Exercise 3 Given the SDE

dXt = (6 − X^t)dt + 6dW^t with Xt=0 = X0 ∼ N (0, 4)

a find the solution

b find E(X^t) and V ar(X^t) and study their behavior when t → ∞ Exercise 4 Suppose you have a SDE of the form

dX^t= (aXtⁿ+ bX^t)dt + cX^tdW^t

prove that the substitution y = h(x) = x¹⁻ⁿ reduces the SDE to a linear SDE with multiplicative noise.

Exercise 5 Give the definition of Diffusion process and discuss an example.

1