# Reflected SDE with non-Lipschitz coefficients

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!

• +1 Interesting question. However, as remarked in mathoverflow.net/questions/249126/… there is no clear advantage to writing the SDE coefficients that way. Therefore, can you please write this SDE in the conventional way and then state your hypotheses on the coefficients? – Nawaf Bou-Rabee Sep 7 '16 at 20:00
• Point taken, I edited the post. – Pcw. Sep 7 '16 at 20:33

## 1 Answer

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.

• Great, that seems very promising. It will take me a while for my untrained ears to grasp it.Thanks! – Pcw. Sep 8 '16 at 1:38