I have an equation of the form:


$$dX\_t=\mu(X\_t)dt+\sigma(X\_t)dZ\_t+dL\_t, \quad X\_0=x\_0\in (-\infty,a]$$


where, $L\_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow the process to get past a barrier $a>0$. 


While I was able to find many results concerning the structure I must impose on the coefficients $\mu(\cdot)$ and $\sigma(\cdot)$ to get strong solutions, pathwise uniqueness, weak solutions and so on for processes without a reflecting barrier, I was unable to find much on processes with reflection. 


In particular, I don't want to assume that $\mu(\cdot)$ and $\sigma(\cdot)$ are Lipschitz functions, but instead, I want to allow it to have a finite number of discontinuities. It would be nice if there was some result on strong solutions for that case (or at least weak solutions with pathwise uniqueness and/or uniqueness in law). 


I am aware of "Dupuis, Paul, and Hitoshi Ishii. SDEs with oblique reflection on nonsmooth domains", but I think it does not cover my case (maybe it does for weak solutions, but there is nothing about uniqueness results or strong solutions).


Thanks!



Reflected SDE with non-Lipschitz coefficients
The given SDE is a special case of a one-dimensional SDE with a **drift measure**
$$
dX = \int\_{\mathbb{R}} d \Lambda\_X(t,x) \mu(dx) + \sigma(X(t)) dZ(t) \tag{$\star$}
$$
where from left: $\Lambda\_Y(t,x)$ is the (symmetric) local time of $X(t)$ at the level $x$, $\mu$ is a measure that we will specify shortly, $\sigma$ is a measurable positive function, and $Z$ is the OP's notation for Brownian motion. For example, ($\star$) reduces to a standard one-dimensional SDE (without local time terms) when the measure $\mu$ is absolutely continuous with respect to Lebesgue measure and $d \mu / d \lambda = b/\sigma^2$ where $\lambda$ is Lebesgue measure on $\mathbb{R}$. 


For simplicity, assume that the singular continuous part of the measure $\mu$ is zero and write
$$
\mu(dx) = \phi(x) \lambda(dx) + \sum\_{i} (2 a\_i - 1) \delta\_{x\_i} (dx) 
$$
where $\phi$ is assumed to be measurable. Basically, we decomposed $\mu$ into a part that is absolutely continuous with respect to Lebesgue measure and another part which is discrete, i.e., a (countable) sum of point masses. Substituting this decomposition back into ($\star$) gives the SDE:
$$
dX = \phi \sigma^2 dt + \sum\_{i} (2 a\_i - 1) d \Lambda\_X(t,x\_i) + \sigma(X(t)) dZ(t) \tag{$\star \star$}
$$
Please note that ($\star \star$) is allowed to have countable number of local time terms. In this general context, strong existence and uniqueness was proven by J.- F. Le Gall (1984); see below for a detailed citation. I included a few more related works, which might be useful. To be sure, the SDE given by the OP is a special case of ($\star \star$) with a single local time term at $x\_0 = a$ and $a\_0 = 1$.


**References**


*Strong Existence and Uniqueness Result*


Le Gall, J.-F. **One-dimensional stochastic differential equations involving the local times of the unknown process.** Stochastic analysis and applications (Swansea, 1983), 51–82, Lecture Notes in Math., 1095, Springer, Berlin, 1984.


*Closely Related Works*


Bass, Richard F., and Zhen-Qing Chen. **One-dimensional stochastic differential equations with singular and degenerate coefficients.** Sankhyā: The Indian Journal of Statistics (2005): 19-45.
APA 


Lejay, Antoine, and Miguel Martinez. **A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients.** The Annals of Applied Probability (2006): 107-139.


Étoré, Pierre. **On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients.** Electron. J. Probab 11.9 (2006): 249-275.


