Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Question

Suppose that $X(t)$ satisfies

$\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$

with $X(0)=0.7.$ Find the mean and the variance of $X(1).$

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I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in variance. I am not sure how to calculate it. Can I use Ito's formula or some other method will used to solve this? Any help will be appreciated. Thanks

Answer 1

Yes, apply Ito to $y=f(x)=x^2$ to get for $Y_t=f(X_t)$ the SDE $$dY_t=a(t,Y_t)\,dt+b(t,Y_t)\,dW_t.$$ Then with $\mu_1(t)=E[X_t]$ and $\mu_2(t)=E[X_t^2]$ you should be able to justify the differential equations $$ \mu_1'(t)=-1.5\mu_1(t)\\ \mu_2'(t)=a(t,\mu_2(t)) $$ where the $t$-dependence of $a$ can include terms with $\mu_1(t)$. Note that the variance of $X_t$ is $\mu_2(t)-\mu_1(t)^2$.