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

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 Hölder 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.

27 pages, corresponds to Part 2 of our previous article [arXiv:1112.4824v1]; to appear in Transactions of the American Mathematical Society

References in corpus (6)

• Mimicking an Itô process by a solution of a stochastic differential equation
• A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients
• Mimicking the marginal distributions of a semimartingale
• Maximum principles for boundary-degenerate second-order linear elliptic differential operators
• Localizing Volatilities
• Maximum principles for boundary-degenerate linear parabolic differential operators

Cited by in corpus (4)

• A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients
• C^{1,1} regularity for degenerate elliptic obstacle problems
• Heat Kernels, Solvable Lie Groups, and the Mean Reverting SABR Stochastic Volatility Model
• Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions