The Martingale Problem Method Revisited

TL;DR

This paper revisits the martingale problem method to establish convergence criteria for a broad class of stochastic processes, including those with discontinuities, extending previous frameworks and deriving new results.

Contribution

It generalizes the martingale problem approach to non-Markovian and discontinuous processes, providing new convergence criteria and theoretical insights.

Findings

Extended martingale problem framework to processes with fixed discontinuities

Derived new convergence criteria for non-Markovian processes

Generalized known results to broader classes of stochastic processes

Abstract

We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales), nor to continuous or cadlag paths. We illustrate our findings both, by finding generalizations of known results, and proving new results. For the latter, we work on processes with fixed times of discontinuity.