# Existence/Uniqueness of the solutions to SDEs of locally Lipschitz coefficients

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.

• The following results should answer your question: (1) existence of strong solution for locally Lipschitz coefficients (with possible blow-ups): see Chapter IV Theorem 3.1 of Ikeda and Watanabe, (2) existence of strong solution without blow-ups for less than locally Lipschitz coeffs but with an additional weakly coercive assumption: see Theorem 3.1.1 of Prévot and Rockner, "A Concise Course on Stochastic Partial Differential Equations" Oct 19, 2022 at 12:47

## 2 Answers

As I cannot upload a screenshot in comments, let me write my answer here.

This is not a complete answer but allows to ensure the existence/uniqueness result. Your question is discussed in pages 134-135 of Numerical Solution of Stochastic Differential Equations, see the screenshot as below :

• Thanks a lot for the answer, while the uniqueness is still not clear to me. Indeed, it seems that the authors believe that the uniqueness is not ensured only by the local Lipschitz continuity. However, the example $dX_t=|X_t|^{\alpha}dW_t$, the function $|x|^{\alpha}$ is not local Lipschitz near zero. So, what is the conclusion? Does the local Lipschitz continuity yield the uniqueness? Nov 17, 2021 at 11:34

Like for ordinary ODEs, SDEs with only locally Lipschitz coefficients may blow-up at a finite finite.

So you need to define well-posedness "up to blow-up time".

It seems to me that if you do that you will obtain strong well-posedness.

Existence may be obtained simply by taking the limit of solutions stopped when exiting a large ball of size $$n$$ and then let $$n \to +\infty$$.

Uniqueness may be obtained by showing that two solutions stopped when exiting a large ball of size $$n$$ are the same, by usual techniques, or by modifiying the coefficients outside the ball to make them globally Lipschitz.

This is a guess, needs to be double checked.

• Thanks for the reply. I do think so, which is also what the paragraph above claims. I wish to find a rigorous argument (theorem or proposition), especially for the uniqueness result. Anyway, I'm a bit surprised that this results, apearing importnat and natural, can not be found in a lot of well known books... Nov 17, 2021 at 13:03