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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!

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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. The Annals of Probability (1993): 554-580.

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  • $\begingroup$ Thank you very much, again. I am still trying to understand that paper (and it has been hard given my poor training on stochastic calculus). If I understood correctly, to guarantee strong uniqueness Dupuis and Ishii assume a Lipschitz condition on the coefficients σ and μ (Corollary 5.2). Without the Lipschitz conditions (and under some other assumptions on the domain and the direction), there is a weak solution. That is already useful, but it would be nice to have a stronger result without the Lipschitz conditions (uniqueness in law, or pathwise uniqueness, for instance) $\endgroup$ – Pcw. Sep 7 '16 at 17:26
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    $\begingroup$ Can we resolve your previous question first? $\endgroup$ – Nawaf Bou-Rabee Sep 7 '16 at 18:20
  • $\begingroup$ I am sorry, I realized now it is a different question from my original one, which was more general. To keep things organized, I created a more specific question here: mathoverflow.net/questions/249286/… $\endgroup$ – Pcw. Sep 7 '16 at 19:08

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