(Moment-generating function for Cox-Ingersoll-Ross process).
(i) Let $W_1, \ldots, W_d$ be independent Brownian motions and let $a$ and $\sigma$ be positive constants. For $j=1, \ldots, d$, let $X_j(t)$ be the solution of the OrnsteinUhlenbeck stochastic differential equation
6.9 Exercises
287
$$
d X_j(t)=-\frac{b}{2} X_j(t) d t+\frac{1}{2} \sigma d W_j(t) .
$$
Show that
$$
X_j(t)=e^{-\frac{1}{2} b t}\left[X_j(0)+\frac{\sigma}{2} \int_0^t e^{\frac{1}{2} b u} d W_j(u)\right] .
$$
Show further that for fixed $t$, the random variable $X_j(t)$ is normal with
$$
\mathbf{E} X_j(t)=e^{-\frac{1}{2} b t} X_j(0), \quad \operatorname{Var}\left(X_j(t)\right)=\frac{\sigma^2}{4 b}\left[1-e^{-b t}\right]
$$