I look for references on the existence/uniqueness of the solution to SDE

$$dX_t = b(t,X_t)dt + a(t,X_t)dW_t,\quad \forall t\ge 0,$$

where $b :\mathbb R_+\times\mathbb R\to\mathbb R$, $a :\mathbb R_+\times\mathbb R\to\mathbb R_+$ are locally Lipschitz, i.e. for any $R>0$, $b :[0,R]\times [-R,R]\to\mathbb R$, $b :[0,R]\times [-R,R]\to\mathbb R_+$ are $L_R-$Lipschitz.

My question is , for any $X_0=x\in\mathbb R$, does the above SDE admits a unique strong solution? I have checked the book Multidimensional Diffusion Processes and Brownian Motion and Stochastic Calculus , but do not find related results.

Any answer, comments and references are highly appreciated.

PS : This question comes to me when I consider the absorbing points of $dX_t=(1-X_t^2)dW_t$. $-1, 1$ are the absorbing points if $dX_t=(1-X_t^2)dW_t$ has a unique (strong) solution. Surprisingly, even this SDE seems to have very simple structure, the classical results cannont be used directly.