Ornstein–Uhlenbeck stochastic differential equation help

Question

Consider the Ornstein–Uhlenbeck stochastic differential equation: $$dX_t =−aX_tdt+σdW_t, X_0 =x_0 ∈ R$$ where $a$ and $σ$ are constants, and $W = (W_t)$, $t≥0$ is a Brownian motion.

i) Using Itô’s formula for the process $e^{at}X_t$ verify that $X_t=X_0e^{-at}+\int\limits_o^t e^{-a(t-s)}dW_s$ and compute $E[X_t]$ and $Var(X_t)$

ii) Derive a stochastic differential equation for the process $Y=(Y_t)_{t≥0}$, where $Y_t = X^{2}_t$

I have managed to complete part i) but I have no idea how to complete part ii). Would I do a similar method to part i), but instead use Itô's formula for the process $e^{at}X^{2}_t$?

Thank you

Answer 1

No, just use Itô's formula again for $Y_t = f(X_t,t) = X_{t}^{2}$. Then \begin{equation} dY_t = \dfrac{\sigma^2}{2}\dfrac{\partial^2 f}{\partial x^2}(X_t,t) dt + \dfrac{\partial f}{\partial x}(X_t,t)dX_t = \\ \left( \sigma^2 - 2aY_{t} \right)dt + 2\sigma \sqrt{Y_t} dW_t. \end{equation}