A new formula for some linear stochastic equations with applications

Details A new formula for some linear stochastic equations with applications A new formula for some linear stochastic equations with applications

2010-09-17

arxiv.org/abs/1009.3373v1

Annals of Applied Probability 2010, Vol. 20, No. 2, 367-381

Mathematics

Probability

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

Scientific paper

10.1214/09-AAP637

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.

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