how to solve SDE

Question

How to solve this SDE: $dX(t)=X^\alpha(t)dt+\sigma X(t)dW(t), for X(0)=x_0$
$dW(t)$ is Wiener process. Also I have to use $f(t)=X(t)exp(-\alpha W(t)+1/2\alpha^2t)$ as integration factor. I tried to solve it looking at solutions of other SDE, but can't find the way for solving non-linear one as I started studying SDE just recently.

The only idea I have is to multiply both sides by integration factor. But what I can do with non-linear part?

What is $X^α(t)$? Is it $X(t)^α$ or some input function with parameter $α$ or some other $α$ modification of $X$ (stopped process?)? — Lutz Lehmann, Jun 29 '17 at 05:50

score 0 · Accepted Answer · answered Jun 29 '17 at 06:22

A la Bernoulli you get for $Y=X^{1-α}$ $$ dY_t=(1-α)X_t^{-α}dX_t+\tfrac12(1-α)(-α)X_t^{-α-1}d\langle X\rangle_t \\ =(1-α)(dt+σY_tdW_t)-\tfrac12α(1-α)σ^2Y_tdt \\ =(1-α)\Big(1-\tfrac12ασ^2Y_t\Big)dt+(1-α)σY_tdW_t $$ which is now linear in $Y$.

how to solve SDE

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