A new formula for some linear stochastic equations with applications

Offer Kella*, Marc Yor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageAmerican English
Pages (from-to)367-381
Number of pages15
JournalAnnals of Applied Probability
Issue number2
StatePublished - Apr 2010


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


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