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?
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.
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.
