Expectation of stochastic differential equation

Question

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito formula). The solution reads $X_{t}=(X_{0}^{1/a}+W_{t})^a$, however I'm stuck on how to compute the expectation $E[X_{t}]$. I am new to stochastic analysis so any kind of help is greatly appreciated! Thanks

Answer 1

The distribution of $X_0$ should be given by some initial condition, typically it is a constant, but not necessarily and $W_t \sim \mathcal{N}(0,t)$ (i.e. the variance is $t$).

Then once both distributions are known, you have to take the integral.

An alternative approach, when the solution to the SDE is not easily derivable, is to take expected value of both sides of the original SDE, which kills the stochastic part: $$ d\mathbb{E}[X_t] = \frac{a(a-1)}{2} \mathbb{E}[X_t^{1-2/a}]dt $$ which is an ordinary differential equation.