Solving an Ito Diffusion and finding the Expectation/Variance

Question

I have an example in which $Y_t$ is the Ito diffusion with generator $A(f )(x) = αxf '(x) + 2x^2f ''(x).$

Assume $Y_0 = y ∈ R^+.$

I would like to find $Y_t$ and from that would like to find $E [Y_t ]$ and $Var [Y_t ]$.

Can someone please walk me through this example?

Answer 1

The dynamics is given by \begin{align*} dX_t = \alpha X_t dt + 2X_t dW_t. \end{align*} That is, $X_t$ is log-normal. The computation of the expectation and variance is then straightforward.