A new formula for some linear stochastic equations with applications

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 cádlág semimartingale and $Y$ is a cádlág 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 Lévy processes, the resulting $X$ is called a generalized Ornstein-Uhlenbeck process.

Published in at http://dx.doi.org/10.1214/09-AAP637 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Exponential functionals of Levy processes
• Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein--Uhlenbeck processes