A new formula for some linear stochastic equations with applications

TL;DR

This paper introduces a new explicit formula for solutions to certain linear stochastic equations involving semimartingales, with applications to processes like shot-noise, growth-collapse, and generalized Ornstein-Uhlenbeck processes.

Contribution

It provides a novel representation formula for solutions of linear stochastic equations with semimartingale inputs, extending understanding of processes like shot-noise and Ornstein-Uhlenbeck.

Findings

Derived a new solution representation for linear stochastic equations.

Applied the formula to processes with stationary increments and bounded jumps.

Connected the results to well-known processes like shot-noise and Ornstein-Uhlenbeck.

Abstract

We give a representation of the solution for a stochastic linear equation of the form $X_t=Y_t+\int_{(0,t]}X_{s-} \mathrm {d}{Z}_s$ where $Z$ is a c\'adl\'ag semimartingale and $Y$ is a c\'adl\'ag adapted process with bounded variation on finite intervals. As an application we study the case where $Y$ and $-Z$ are nondecreasing, jointly have stationary increments and the jumps of $-Z$ are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When $Y$ and $Z$ are, in addition, independent L\'evy processes, the resulting $X$ is called a generalized Ornstein-Uhlenbeck process.