A quintuple law for Markov additive processes with phase-type jumps

We consider a Markov 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. A marginal 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.

[1]  Guy Latouche,et al.  Risk processes analyzed as fluid queues , 2005 .

[2]  José Garrido,et al.  The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion , 2005 .

[3]  Chuancun Yin,et al.  The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion , 2003 .

[4]  Soohan Ahn,et al.  On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals , 2007 .

[5]  Søren Asmussen,et al.  Ruin probabilities , 2001, Advanced series on statistical science and applied probability.

[6]  Søren Asmussen,et al.  Ladder heights and the Markov-modulated M/G/1 queue☆ , 1991 .

[7]  Zbigniew Palmowski,et al.  Fluctuations of spectrally negative Markov additive processes , 2008 .

[8]  Hans U. Gerber,et al.  On the discounted penalty at ruin in a jump-diffusion and the perpetual put option , 1998 .

[9]  Guy Latouche,et al.  The surplus prior to ruin and the deficit at ruin for a correlated risk process , 2005 .

[10]  Hailiang Yang,et al.  On the joint distribution of surplus before and after ruin under a Markovian regime switching model , 2006 .

[11]  Lothar Breuer.  First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type , 2008, Journal of Applied Probability.

[12]  B. A. Rogozin,et al.  On Joint Distributions of Random Variables Associated with Fluctuations of a Process with Independent Increments , 1969 .

[13]  Hans U. Gerber Asa,et al.  The Time Value of Ruin in a Sparre Andersen Model , 2005 .

[14]  “On the Expected Discounted Penalty Function for L´vy Risk Processes,” José Garrido and Manuel Morales, October 2006 , 2007 .

[15]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[16]  José Garrido,et al.  On The Expected Discounted Penalty function for Lévy Risk Processes , 2006 .

[17]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[18]  David M. Lucantoni,et al.  New results for the single server queue with a batch Markovian arrival process , 1991 .

[19]  A Note on Wiener–Hopf Factorization for Markov Additive Processes , 2009, 0906.1223.

[20]  A. E. Kyprianou,et al.  Overshoots and undershoots of Lèvy processes , 2006 .

[21]  M. Neuts.  A Versatile Markovian Point Process , 1979 .

[22]  S. Asmussen,et al.  Marked point processes as limits of Markovian arrival streams , 1993 .

[23]  M. Pistorius,et al.  ON MAXIMA AND LADDER PROCESSES FOR A DENSE CLASS OF LEVY PROCESS , 2005 .

[24]  S. Asmussen,et al.  Russian and American put options under exponential phase-type Lévy models , 2004 .

[25]  Yi Lu,et al.  The Expected Discounted Penalty at Ruin for a Markov-Modulated Risk Process Perturbed by Diffusion , 2007 .