The Annals of Applied Probability

A new formula for some linear stochastic equations with applications

Offer Kella and Marc Yor

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We give a representation of the solution for a stochastic linear equation of the form Xt=Yt+(0, t]Xs dZs where Z is a càdlàg semimartingale and Y is a càdlàg 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.

Article information

Ann. Appl. Probab., Volume 20, Number 2 (2010), 367-381.

First available in Project Euclid: 9 March 2010

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Zentralblatt MATH identifier

Primary: 60H20: Stochastic integral equations
Secondary: 60G51: Processes with independent increments; Lévy processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Linear stochastic equation growth collapse process risk process shot-noise process generalized Ornstein–Uhlenbeck process


Kella, Offer; Yor, Marc. A new formula for some linear stochastic equations with applications. Ann. Appl. Probab. 20 (2010), no. 2, 367--381. doi:10.1214/09-AAP637.

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