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

Question

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！

Answer 1

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).