A Dirichlet Process Characterization of a Class of Reflected Diffusions

For a general class of stochastic differential equations with reflection that admit a Markov weak solution and satisfy a certain L continuity condition, p > 1, it is shown that the associated reflected diffusion can be decomposed as the sum of a local martingale and a continuous, adapted process of zero p-variation. In particular, when p = 2, this implies that the associated reflected diffusion is a Dirichlet processes in the sense of Föllmer. As motivation for such a characterization, it is also shown that reflected diffusions belonging to a specific family within this class are not semimartingales, but are Dirichlet processes. This family of diffusions arise naturally as approximations of certain stochastic networks that use the so-called generalized processor sharing scheduling policy.