A new formula for some linear stochastic equations with applications
| Autor: | Marc Yor, Offer Kella |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2010 |
| Předmět: |
Statistics and Probability
01 natural sciences Lévy process Linear stochastic equation 010104 statistics & probability growth collapse process FOS: Mathematics Applied mathematics 0101 mathematics Representation (mathematics) 60H20 ComputingMilieux_MISCELLANEOUS generalized Ornstein–Uhlenbeck process Mathematics risk process shot-noise process Probability (math.PR) 010102 general mathematics Multiplicative function Adapted process [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] Semimartingale Bounded function Bounded variation 60K30 Statistics Probability and Uncertainty 60G51 Linear equation Mathematics - Probability |
| Zdroj: | Annals of Applied Probability Annals of Applied Probability, 2010, 20 (2), pp.367-381. ⟨10.1214/09-AAP637⟩ Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2010, 20 (2), pp.367-381 Ann. Appl. Probab. 20, no. 2 (2010), 367-381 |
| ISSN: | 1050-5164 2168-8737 |
| DOI: | 10.1214/09-AAP637⟩ |
| Popis: | 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. Comment: 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) |
| Databáze: | OpenAIRE |
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