A Quintuple Law for Markov Additive Processes with Phase-type Jumps
نویسنده
چکیده
Weconsider aMarkov additive process (MAP)with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. Amarginal of the derived distribution is the Gerber–Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.
منابع مشابه
First Passage times for Markov Additive Processes with Positive Jumps of Phase Type
The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numer...
متن کاملExit Problems for Reflected Markov-additive Processes with Phase–type Jumps
Let (X ,J ) denote a Markov-additive process with phase–type jumps (PH-MAP) and denote its supremum process by S. For some a > 0, let τ(a) denote the time when the reflected processY := S−X first surpasses the level a. Further, let τ−(a) denote the last time before τ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of Sτ(a), τ−(a) and τ(a), where the ...
متن کاملOn maxima and ladder processes for a dense class of Lévy processes
Consider the problem to explicitly calculate the law of the first passage time T (a) of a general Lévy process Z above a positive level a. In this paper it is shown that the law of T (a) can be approximated arbitrarily closely by the laws of T (a), the corresponding first passages time for X, where (X)n is a sequence of Lévy processes whose positive jumps follow a phase-type distribution. Subse...
متن کاملEXACT AND ASYMPTOTIC n - TUPLE LAWS AT FIRST AND LAST PASSAGE
Understanding the space–time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) ...
متن کاملThe First Passage Event for Sums of Dependent Lévy Processes with Applications to Insurance Risk by Irmingard
For the sum process X =X1 +X2 of a bivariate Lévy process (X1,X2) with possibly dependent components, we derive a quintuple law describing the first upwards passage event of X over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undersho...
متن کامل