A note on time-dependent additive functionals

TL;DR

This paper extends the theory of time-inhomogeneous additive functionals, providing tools for analyzing time-dependent Markov processes and their applications in backward stochastic differential equations, especially concerning quadratic variation and martingale properties.

Contribution

It introduces extensions of key properties of additive functionals to non-homogeneous Markov processes, aiding the analysis of time-dependent stochastic models.

Findings

Extended quadratic variation results to non-homogeneous settings

Provided new insights into martingale additive functionals in time-inhomogeneous processes

Supported analysis of BSDEs in law with these extended tools

Abstract

This note develops shortly the theory of time-inhomogeneous additive functionals and is a useful support for the analysis of time-dependent Markov processes and related topics. It is a significant tool for the analysis of BSDEs in law. In particular we extend to a non-homogeneous setup some results concerning the quadratic variation and the angular bracket of Martin-gale Additive Functionals (in short MAF) associated to a homogeneous Markov processes.