Mean and Variance of SDE

Question

How would I compute the mean and variance of the following SDE?

$dX_t = \alpha X_t dt + \sigma dB_t$

I know $E[X_t]$ produces the mean and $E[(X_t)^2]$ produces the variance, but I'm not sure how to do these expectations. Must I use Ito's formula?

Answer 1

Note that \begin{align*} X_t = X_0 e^{\alpha t} + \sigma\int_0^t e^{\alpha(t-s)} dW_s. \end{align*} Therefore \begin{align*} E(X_t) = X_0 e^{\alpha t}, \end{align*} and \begin{align*} E\left(X_t^2\right) &= E\left(\left(X_t-E(X_t)\right)^2\right) + \left(E\left(X_t\right) \right)^2\\ &=\sigma^2 \int_0^te^{2\alpha(t-s)} ds + X_0^2 e^{2\alpha t}\\ &=\sigma^2\left(\frac{1}{2\alpha} e^{2\alpha t}-1 \right) + X_0^2 e^{2\alpha t}. \end{align*}