Stochastic Mechanics 6 CFU Part II 15.7.2009 Exercise 1 Let W

Stochastic Mechanics 6 CFU Part II 15.7.2009

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

a d(tXt²ⁿ) for n ≥ 1, where dX^t= 2dt + t²dW^t b dX^t where X^t= e^t−Wt

c R^T

0 (Wt⁴+ W^t)dW^t

Exercise 2 Given the equation

˙x = λx − x³

a randomize the parameter λ → λ + σξ^t where ξ^t is a white noise and write the corresponding SDE

b solve the SDE (use the substitution y = 1/x²) Exercise 3 Given the SDE

dX^t = (1 + βX^t)dt + dW^t

with Xt=0 = X0 ∼ N (0, 9) and β ∈ R, find E(X^t) and σ²(X^t) and study their behavior as functions of β.

Exercise 4 Given the process X^t find the SDE that it solves in the following cases

a X^t= _1+2t^Wt b X^t= e^3Wt−t

Exercise 5 Write the Kolmogorov forward equation for Brownian mo- tion.

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