# How to solve the SDE $\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t)$?

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.4, the author claims that the problem $$\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t),X(0)=x_0$$ has a unique solution for all $$t>0$$. Here $$B(t)$$ is Brownian motion. But up to this point, the book has not mentioned uniqueness theorem. Hence, I think that may be the SDE can be solved. But I don't know how to solve it?

Also, the book gives only one hint that since the drift coefficient directs the motion to the origin, the solution of the SDE is unique. But I don't understand it. Could you give me more hints?

Thanks！

• This answer shows what an arduous task it can be to "solve" such non linear SDEs. Existence and uniqueness theorems in the spirit of Picard-Lindelöf (from ODEs) exist for SDEs as well and a book by Khasminskii cannot be that bad to just give the hint that the direction of the drift coefficient guarantees uniqueness. Keep reading about those standard theorems. Commented Apr 17 at 8:29

As mentioned here in 2.2.3 Theorem, one simply needs locally Lipschitz to get uniqueness (existence needs additional conditions such as linear growth)

Suppose that $$b$$ and $$\sigma$$ are locally Lipschitz continuous in the space variable, that is, for all $$n \in N$$ there is a $$K_n > 0$$ such that for all $$t \geq 0$$ and all $$x, y \in R^d$$ with $$|x|,|y|\leq n$$ $$|b(x,t)-b(y,t)|+|\sigma(x,t)-\sigma(y,t)|\leq K_n |x-y|$$ holds. Then strong uniqueness holds.

Another quick way to see uniqueness is to study the difference $$D(t)=X(t)-Y(t)$$. We have

$$(D^2)'=(X-Y)(Y^3-X^3)\leq 0$$

and so $$D^2=0$$.

By the way, this SDE is also a type of Langevin-SDE (see details here Steady state density of Langevin SDE on how to study their long term behaviour).

• How can we get $D^2=0$ from $D'<0$? Thanks! Commented Sep 5 at 8:42
• @R-CH2OH Since $D^{2}(0)=0$, we have $$0\leq D^2(t)=\int_{0}^{t}(D^{2}(s))'ds\leq 0.$$ Commented Sep 5 at 15:29