Abstract
Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $ D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.
Highlights
There are many ways of specifying Markov processes, the most popular being as solutions of stochastic equations, as solutions of martingale problems, or in terms of solutions of the Kolmogorov forward equation
The solution of a stochastic equation explicitly gives a process while a solution of a martingale problem gives the distribution of a process and a solution of a forward equation gives the one dimensional distributions of a process
One approach to proving uniqueness for a forward equation and for the corresponding martingale problem is to verify a condition on the generator similar to the range condition of the Hille-Yosida theorem. (See Corollary 2.14.) We show that the original generator A of our martingale problem can be extended to a generator A such that every solution of the martingale problem for A is a solution for A and A satisfies the range condition of Corollary 2.14
Summary
There are many ways of specifying Markov processes, the most popular being as solutions of stochastic equations, as solutions of martingale problems, or in terms of solutions of the Kolmogorov forward equation (the Fokker-Planck equation or the master equation depending on context). We offer an abstract definition of viscosity sub/supersolution for (1.1) (which for integro-differential operators in Rd is equivalent to the usual one) and prove, under very general conditions, that the martingale problem for A has a unique solution if the comparison principle for (1.1) holds. Processes in domains with boundaries that are only piecewise smooth or with boundary operators that are second order or with directions of reflection that are tangential on some part of the boundary continue to be a challenge In this last case, as an example of an application of our results, we use the comparison principle proved in [25] to obtain uniqueness. The rest of this paper is organized as follows: Section 2 contains some background material on martingale problems and on viscosity solutions; Section 3 deals with martingale problems; the alternative definitions of viscosity solution are discussed in Section 4; Section 5 deals with martingale problems with boundary conditions; in Section 6, we present two examples, including the application to diffusions with degenerate direction of reflection
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