Path-dependent Poisson random measures and stochastic integrals constructed from general point processes

TL;DR

This paper introduces a novel path-dependent Poisson random measure called Mesgaki measure, extending traditional models to incorporate path-dependent jump characteristics, with applications in continuous-time reinforcement learning.

Contribution

It constructs a new class of Poisson random measures from general point processes, including their stochastic integrals and Itô's formula, advancing the mathematical framework for path-dependent stochastic modeling.

Findings

Defined Mesgaki random measure as a limit of counting processes

Developed stochastic integral and Itô's formula for the measure

Extended Poisson random measures to include path-dependent features

Abstract

In this paper, we consider an extension of the Poisson random measure for the formulation of continuous-time reinforcement learning, such that both the frequency and the width of the jumps depend on the path. Starting from a general point process, we define a new Poisson random measure as limit of the linear sum of these counting processes, and name it the Mesgaki random measure. We also construct its Stochastic integral and It\^o's formula.