The stochastic process {X(t), t 0} is a homogeneous Gaussian process in relation to time

and has the infinitesimal parameters: mX(x) 0 and vX(x) 1 (in order for {X(t), t 0}

to be a Wiener process).

(a) Suppose that X(0) is a random variable uniformly distributed in the interval [1, 1].

Find V [X(t)] for t 0.

(b) We define Y (t) = t

2X(t) for t > 0. Is the process {Y (t), t > 0} Gaussian? Justify

and find the function fY (y;t) if X(0) = 0.

(c) With Z(t) := X3

(t) for t 0. Find the infinitesimal parameters of the process

{Z(t), t 0

Homework 2 Question 1 The stochastic process {X(t), t 0. Is the process {Y(t), t > 0} Gaussian? Justify and find the function fy(y; t) if X (0) = 0. (c) With Z(t) := X3(t) for t 2 0. Find the infinitesimal parameters of the process {Z (t) , t2 0)