I have an equation of the form:


$$dX\_t=\mu(X\_t)X\_tdt+\sigma(X\_t)X\_tdZ\_t+dL\_t, \quad X\_0=x\_0\in (0,a]$$


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


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. 


Does anyone know a good reference for this kind of processes?


Thanks!



Existence of solution for reflected SDE
Sufficient conditions for strong existence and uniqueness of solutions to stochastic differential equations with reflection were derived by Dupuis & Ishii (see below for detailed reference) in the following cases: 


1. the reflection direction is single-valued and varies smoothly, but the boundary of the domain may be nonsmooth; and,
2. the domain of the SDE is the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points there may be more than one reflection direction.


**Reference**


Dupuis, Paul, and Hitoshi Ishii. [SDEs with oblique reflection on nonsmooth domains.](https://projecteuclid.org/euclid.aop/1176989415) The Annals of Probability (1993): 554-580.


