# Solving a Stochastic Differential Equation (SDE)

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

• Note that your result gives $(X_t)$ as a function of itself. Also, $\log$ is not $C^2$ on $\mathbb R_+$ so you have to be careful when applying Itô's formula to $\log$ when you don't know if $(X_t)$ hits $0$ or not. – Ian May 21 '14 at 0:53
• So, could I use another function instead? – dimebucker May 21 '14 at 0:57
• Usually, you either consider the process stopped at $T_0$ the hitting time of zero, and then check that $T_0=+\infty$ almost surely, or you solve "as if it were $C^2$", and then check that the derived solution indeed solves the SDE. Aside from this technicality, you have proved that considering $\log$ does not help you derive a solution in this case. – Ian May 21 '14 at 0:59

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 $t$.
• $$F(W_t)=\frac 1{W_t}$$but why those conditions? – dimebucker May 21 '14 at 7:34
• You want $X$ to solve the SDE $dX=X^3dt-X^2dW$. Hence, if $dX=F'(W)dW+\frac12F''(W)dt$, you need that... // Note that $1/W$ is not defined at time $0$ and that you need $F(0)=1$ (as written in my answer) hence no, $F(w)=1/w$ is not a solution. – Did May 21 '14 at 7:37
• thanks, this makes sense now, I get that $F(W_t)=\frac {1}{W_t+1}$ – dimebucker May 29 '14 at 13:39