A representation formula for the probability density in stochastic dynamical systems with memory

TL;DR

This paper develops a novel representation formula for the probability density of Marcus stochastic delay differential equations (SDDEs) by relating them to non-delay Marcus SDEs, overcoming the lack of traditional Fokker-Planck equations.

Contribution

It introduces a method to express the density of Marcus SDDEs using Marcus SDEs without delays, enabling analysis despite non-Markovian properties.

Findings

Provides a representation formula for the density of Marcus SDDEs.

Establishes existence and uniqueness of solutions for these SDDEs.

Links the density of SDDEs to that of non-delay Marcus SDEs.

Abstract

Marcus stochastic delay differential equations (SDDEs) are often used to model stochastic dynamical systems with memory in science and engineering. Since no infinitesimal generators exist for Marcus SDDEs due to the non-Markovian property, conventional Fokker-Planck equations, which govern the evolution behavior of density, are not available for Marcus SDDEs. In this paper, we identify the Marcus SDDE with some Marcus stochastic differential equation (SDE) without delays but subject to extra constraints. This provides an efficient way to establish existence and uniqueness for the solution, and obtain a representation formula for probability density of the Marcus SDDE. In the formula, the probability density for Marcus SDDE is expressed in terms of that for Marcus SDE without delay.