# Locally Feller processes and martingale local problems

## TL;DR

This paper investigates martingale problems related to locally Feller processes with finite lifetime, analyzing properties like weak convergence, Feller features, and localization theorems for well-posed problems.

## Contribution

It introduces a framework for studying martingale problems with finite lifetime processes, emphasizing Feller properties and localization techniques.

## Key findings

- Weak convergence of solutions under Skorokhod topology
- Identification of Feller-type features in solutions
- Localization theorems for well-posed martingale problems

## Abstract

This paper is devoted to the study of a certain type of martingale problems associated to general operators corresponding to processes which have finite lifetime. We analyse several properties and in particular the weak convergence of sequences of solutions for an appropriate Skorokhod topology setting. We point out the Feller-type features of the associated solutions to this type of martingale problem. Then localisation theorems for well-posed martingale problems or for corresponding generators are proved.