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
If $X_t=F(W_t)$, one knows that $\mathrm dX_t=F'(W_t)\mathrm dW_t+\frac12F''(W_t)\mathrm dt$. If ever there exists some function $F$ such that $$F(0)=1,\qquad F'(w)=-F(w)^2,\qquad F''(w)=2F(w)^3,$$ the proof is complete. Can you identify such a function $F$? Be aware though that there might be no solution $(X_t)$ defined for every nonnegative $t$.