TY - JOUR
T1 - A new formula for some linear stochastic equations with applications
AU - Kella, Offer
AU - Yor, Marc
PY - 2010/4
Y1 - 2010/4
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
UR - http://www.scopus.com/inward/record.url?scp=77949737223&partnerID=8YFLogxK
U2 - 10.1214/09-AAP637
DO - 10.1214/09-AAP637
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AN - SCOPUS:77949737223
SN - 1050-5164
VL - 20
SP - 367
EP - 381
JO - Annals of Applied Probability
JF - Annals of Applied Probability
IS - 2
ER -