Unifying the Dynkin and Lebesgue-Stieltjes formulae

TL;DR

This paper introduces a unified local martingale framework that combines Dynkin's formula and Lebesgue-Stieltjes integration, applicable to jump diffusions and general Markov processes, with conditions for $L^2$ martingale properties and convergence.

Contribution

It establishes a novel local martingale that unifies classical formulas for Markov processes and functions of bounded variation, extending their applicability.

Findings

Unified martingale framework for Markov processes and bounded variation functions

Conditions under which the martingale is an $L^2$ martingale

Sufficient conditions for convergence of the martingale to zero

Abstract

We establish a local martingale $M$ associate with $f(X,Y)$ under some restrictions on $f$, where $Y$ is a process of bounded variation (on compact intervals) and either $X$ is a jump diffusion (a special case being a L\'evy process) or $X$ is some general (c\'adl\'ag metric space valued) Markov process. In the latter case $f$ is restricted to the form $f(x,y)=\sum_{k=1}^K\xi_k(x)\eta_k(y)$. This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue-Stieltjes integration (change of variable) formula for (right continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed then this local martingale becomes an $L^2$ martingale. Convergence of the product of this Martingale with some deterministic function (of time) to zero both in $L^2$ and a.s. is also considered and sufficient conditions for functions for which this happens are identified.