On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Ito processes

TL;DR

This paper addresses the well-posedness of the martingale problem and the existence of Markov processes for degenerate diffusion operators with unbounded coefficients, with implications for probability theory and finance.

Contribution

It establishes well-posedness of the martingale problem, existence and uniqueness of solutions to degenerate SDEs, and a mimicking theorem for Ito processes with unbounded coefficients.

Findings

Martingale problem is well-posed for degenerate-elliptic operators with unbounded coefficients.

Existence and uniqueness of weak solutions to degenerate SDEs with strong Markov property.

Existence of a Markov process matching the marginals of a given Ito process.

Abstract

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that the martingale problem associated with a degenerate-elliptic differential operator with unbounded, locally Holder continuous coefficients on a half-space is well-posed in the sense of Stroock and Varadhan. Second, we prove existence, uniqueness, and the strong Markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable H\"older continuity properties. Third, for an Ito process with degenerate diffusion and unbounded but appropriately regular coefficients, we prove existence of a strong Markov process, unique in the sense of probability law, whose one-dimensional marginal probability distributions match those of the given Ito process.