The time to ruin for a class of Markov additive risk process with two-sided jumps.

*(English)*Zbl 1100.60021Ruin problems for a special Markov additive process are considered. The model process locally behaves like a real-valued Brownian motion with some drift and variance, these both depending on an underlying Markov chain that is also used to generate the claims arrival process. Thus, claims arrive according to a renewal process with waiting times of phase type. Positive claims (downward jumps) are always possible but negative claims (upward jumps) are also allowed. The claims are assumed to form an independent, identically distributed sequence, independent of everything else. The time for ruin is the (stopping) time at which the process becomes strictly negative, and the undershoot is the absolute value of the process at the time to ruin.

As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve martingale techniques by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. Moreover, complex function theory is applied.

As main results, the joint Laplace transform of the time to ruin and the undershoot at ruin, as well as the probability of ruin, are explicitly determined under the assumption that the Laplace transform of the positive claims is a rational function. Both the joint Laplace transform and the ruin probability are decomposed according to the type of ruin: ruin by jump or ruin by continuity. The methods used involve martingale techniques by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. Moreover, complex function theory is applied.

Reviewer: Uta Freiberg (Canberra)

##### MSC:

60G40 | Stopping times; optimal stopping problems; gambling theory |

60K15 | Markov renewal processes, semi-Markov processes |

60E10 | Characteristic functions; other transforms |

60G44 | Martingales with continuous parameter |

60J25 | Continuous-time Markov processes on general state spaces |

91B30 | Risk theory, insurance (MSC2010) |

##### Keywords:

Markov process; optional sampling; risk model; martingale; probability of ruin; time to ruin; undershoot; additive process
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\textit{M. Jacobsen}, Adv. Appl. Probab. 37, No. 4, 963--992 (2005; Zbl 1100.60021)

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