Ruin probabilities for a Lévy-driven generalised Ornstein-Uhlenbeck process. (English) Zbl 1430.91031
The paper focuses on the asymptotics of the ruin probability, that is one of the important topics of ruin theory. The authors consider the case of a process which is the solution of a linear SDE defined by a pair of independent Lévy processes,
After some recalls about the Lévy processes, the Authors present the model; then a classic simplification of the problem of ruin by studying the asymptotic behaviour of a stochastic integral is considered.
Moment inequalities for maximal functions of stochastic integrals are given, aiming at the limiting behaviour of an exponential functional.
Finally the authors treat the case in which ruin is imminent for any initial reserve, providing a theorem on ruin with probability one.
Some examples and a detailed Appendix close the paper.
After some recalls about the Lévy processes, the Authors present the model; then a classic simplification of the problem of ruin by studying the asymptotic behaviour of a stochastic integral is considered.
Moment inequalities for maximal functions of stochastic integrals are given, aiming at the limiting behaviour of an exponential functional.
Finally the authors treat the case in which ruin is imminent for any initial reserve, providing a theorem on ruin with probability one.
Some examples and a detailed Appendix close the paper.
Reviewer: Emilia Di Lorenzo (Napoli)
MSC:
| 91B05 | Risk models (general) |
| 60J60 | Diffusion processes |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
ruin probabilities; dual models; price process; renewal theory; distributional equation; autoregression with random coefficients; Lévy processReferences:
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