Question: Solve the stochastic differential equation:
$$ dX_t=X^3_t\,dt-X^2_t\,dW_t $$ where: $$ X_0=1 $$
My Attempt:
Using Ito's with: $$ f(x)=\log(x) $$ I get that: $$ d\log(X_t)=dt\left(0+\left(\frac 1{X_T}\right)X_T^3+\left(\frac {-1}{X_T^2}\right)X^4_T\right)+\left(\frac 1{X_T}\right)\left(-X_T^2\right)\,dW_t $$ which gives me that: $$ d\log X_t=-X_t\,dW_t $$ and integrating from 0 to t: $$ X_t=\exp\left(-\int_0^tX_s\,dW_s\right) $$ Is this correct? I don't have too much experience with SDEs so would appreciate any hints/advice.
Cheers