A new formula for some linear stochastic equations with applications

Offer Kella; Marc Yor

doi:10.1214/09-AAP637

A new formula for some linear stochastic equations with applications

Offer Kella^*, Marc Yor

^*Corresponding author for this work

Department of Statistics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We give a representation of the solution for a stochastic linear equation of the form X_t = Y_t + ∫_{(0, t]} X _s- dZ_s 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.

Original language	English
Pages (from-to)	367-381
Number of pages	15
Journal	Annals of Applied Probability
Volume	20
Issue number	2
DOIs	https://doi.org/10.1214/09-AAP637
State	Published - Apr 2010

Keywords

Generalized Ornstein-Uhlenbeck process
Growth collapse process
Linear stochastic equation
Risk process
Shot-noise process

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10.1214/09-AAP637

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title = "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 Xt = Yt + ∫(0, t] X s- dZs where Z is a c{\`a}dl{\`a}g semimartingale and Y is a c{\`a}dl{\`a}g 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{\'e}vy processes, the resulting X is called a generalized Ornstein-Uhlenbeck process.",

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author = "Offer Kella and Marc Yor",

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T1 - A new formula for some linear stochastic equations with applications

AU - Kella, Offer

AU - Yor, Marc

PY - 2010/4

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N2 - We give a representation of the solution for a stochastic linear equation of the form Xt = Yt + ∫(0, t] X s- 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.

AB - We give a representation of the solution for a stochastic linear equation of the form Xt = Yt + ∫(0, t] X s- 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.

KW - Generalized Ornstein-Uhlenbeck process

KW - Growth collapse process

KW - Linear stochastic equation

KW - Risk process

KW - Shot-noise process

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EP - 381

JO - Annals of Applied Probability

JF - Annals of Applied Probability

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A new formula for some linear stochastic equations with applications

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Keywords

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