Consider the one-dimensional stochastic differential equation:


$$dX\_t = {\bf 1}\_{\{X\_t>0\}}\big(b(t,X\_t)dt + a(t,X\_t)dW\_t\big),\quad \forall t>0,$$


or equivalently


$$dX\_t = b(t,X\_t)dt + a(t,X\_t)dW\_t,\quad \forall t\le \tau,\quad \mbox{with } \tau:=\inf\{t\ge 0: X\_t\le 0\},$$


where $(W\_t)\_{t\ge 0}$ is a standard Brownian motion and $\mathbb P(X\_0>0)=1$. Under which conditions on the coefficients $b, a$ does pathwise uniqueness or even uniqueness in law hold?



Uniqueness of the solution to some degenerate SDE
The solution of the SDE in question is an example of a time-inhomogeneous diffusion process that is stopped/killed/terminated when it first hits the origin. For pathwise uniqueness until time $t \wedge \tau$, Theorem 3.7 in Chapter 5 of the following classic should do the trick.


*Ethier, Stewart N.; Kurtz, Thomas G.*, Markov processes. Characterization and convergence, Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. X, 534 p. \textsterling 49.10 (1986). [ZBL0592.60049](https://zbmath.org/?q=an:0592.60049).


In this setting the theorem precisely states the following.



> 
> **Theorem** (Pathwise Uniqueness). Let $U \subset \mathbb{R}$ be an open set, let $T>0$, and suppose that there exists a constant $L$ such that $$
> |b(t,x)-b(t,y)| \vee |a(t,x)-a(t,y)| \le L |x-y| \qquad 0 \le t \le T \quad x,y \in U \;.$$
> Given two solutions $X\_t, Y\_t$ of the SDE, let $$
> \tau = \inf\{ t \ge 0 \mid X\_t \notin U~~ \text{or}~~ Y\_t \notin U \} \;.
> $$ Then $P[X\_0 = Y\_0] = 1$ implies that $P[X\_{t \wedge \tau} = Y\_{t \wedge \tau}~~\text{for} ~~0 \le t \le T] = 1$.
> 
> 
> 


To be sure, choose the open set to be $U = \{ x > 0 \}$. The proof of this theorem is standard, and involves computing the mean-squared difference between $X\_{t \wedge \tau}$ and $Y\_{t \wedge \tau}$, applying Lipschitz continuity of the coefficients, and then invoking Grönwall's inequality.


