Uniqueness of the solution to stochastic differential equation

Question

Let $W$ be a Brownian motion and consider the SDE

$$dX_t = b(t,X_t) \, dt + a(t,X_t)\,dW_t,\quad \forall t\ge 0. \tag{$\ast$} $$

Assume that $x\mapsto b(t,x), a(t,x)$ are locally Lipschitz in $x$ uniformly in $t$, i.e. for any $T>0$ and $R>0$, there exists $L>0$ s.t.

$$|b(t,x)-b(t,y)|+ |a(t,x)-a(t,y)| \le L|x-y|,\quad \forall 0\le t\le T,~~ |x|, |y|\le R.$$

Then $(\ast)$ admits at most one solution $(X^{s,z})_{t\ge s}$ for any initial condition $X^{s,z}_s=z$. If $X^{0,0}_t\equiv 0$ is a solution to $(\ast)$, do we have

$$\mathbb P[X^{0,z}_t\neq 0, \forall t\ge 0]=1,\quad \forall z\neq 0 \text{?}$$

Any answer, comments and references are highly appreciated.

Answer 1

First for the case of time-independent coefficients (autonomous). A good setting for this question is the Feller-explosion test (see here Is this process strictly positive? for related question). The Feller-explosion test in Karatzas-Shreeve theorem 5.29

So for the case mentioned in the comments, we assume weak solution existence in interval $(0,r)$ and we want to check $X_{t}>0$ for all $t\geq 0$. The Feller-explosion test, says never exiting $(0,b)$ (i.e. $P(S_{(0,b)}=\infty)=1$) is equivalent to a condition on the scale functions. Here

$$v(x):=\int_{x_{0}}^{x}p'(x)\int_{x_{0}}^{y}\frac{2}{p'(z)\sigma^{2}(z)}dz,$$

where $p(x):=\int_{x_{0}}^{x}exp(-2\int_{x_{0}}^{\xi}\frac{b(y)}{\sigma^{2}(y)}dy)d\xi$, for $b$ the drift and $\sigma$ the volatility coefficient.

In the case of time-dependent coefficients (non-autonomous) as in the question $\sigma(t,X_{t})$, there is still an analogue of Feller-explosion i.e. non-escape and thus strictly positive in the work "On the Distribution of Explosion Time of Stochastic Differential Equations.".

where they look at a generalization

It seems the fully non-autonomous case is still open for very general coefficients.