On the Local Times and Boundary Properties of Reflected Diffusions with Jumps in the Positive Orthant

#### F.M. Guillemin, R.R. Mazumdar, F.J. Piera

2006, v.12, №3, 561-582

ABSTRACT

In this paper we study boundary properties of reflected diffusions with positive and negative jumps, constrained to lie in the positive orthant of $R^n$. We consider a model with oblique reflections and characterize the regulator processes in terms of semi-martingale local times at the boundary or reflection faces of $R_{+}^{n}$. In particular, we show that under mild boundary conditions on the diffusion coefficients, and under a completely-S structure for the reflection matrix with an additional invertibility requirement, the regulator processes do not charge the set of times spent by the process at the intersection of two or more boundary faces. Other supporting results are also provided, as for example the fact that the law in $R_{+}^{n}$ of the process at time $t$ does not charge boundary faces for Lebesgue-a.e. $t$. The case of hyper-rectangular state spaces contained in the positive orthant is also considered.

Keywords: reflected diffusions,jumps,oblique reflection directions,regulator processes,local times,semi-martingales