Stochastic Mechanics 6 CFU Part II 13.12.2007 Exercise 1 let W

Stochastic Mechanics 6 CFU Part II 13.12.2007

Exercise 1let Wtbe a Brownian motion. Calculate the Ito differential of the process Xt= e^3Wt

.

Exercise 2The Verhulst equation is used to model population dynamics

˙x = λx − x²

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

bsolve the SDE (use the substitution y = 1/x).

Exercise 3Given the SDE

dX^t= (3 − X^t)dt + 5dW^t with X^t=0= X⁰∼ N (µ, σ²)

afind the solution

bfind E(Xt) and V ar(Xt) and study their behavior when t → ∞.

Exercise 4Give the definition of diffusion process and discuss an example.

Exercise 5Given the SDE

dX^t= −5X^t(1 + Xt²) + 3X^tdW^t afind the steady solution

bprove that it is uniformly stochastically asymptotically stable in the large (use Lyapunov function).

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