Advances in Applied Probability

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

Martin Jacobsen

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We consider risk processes that locally behave like 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. 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 finding certain martingales by first finding partial eigenfunctions for the generator of the Markov process composed of the risk process and the underlying Markov chain. We also use certain results from complex function theory as important tools.

Article information

Adv. in Appl. Probab. Volume 37, Number 4 (2005), 963-992.

First available in Project Euclid: 14 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J25: Continuous-time Markov processes on general state spaces 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60G44: Martingales with continuous parameter 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65]

Probability of ruin time to ruin undershoot passage time martingale optional sampling additive process Rouché's theorem


Jacobsen, Martin. The time to ruin for a class of Markov additive risk process with two-sided jumps. Adv. in Appl. Probab. 37 (2005), no. 4, 963--992. doi:10.1239/aap/1134587749.

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  • Asmussen, S. (2000). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 2). World Scientific, River Edge, NJ.
  • Asmussen, S., Avram, F. and Pistorius, M. R. (2004). Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109, 79--111.
  • Avram, F. and Usábel, M. (2004). Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times. Astin Bull. 34, 315--332.
  • Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215--238.
  • Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156--169.
  • Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, London.
  • Dickson, D. C. M. and Hipp, C. (1998). Ruin probabilities for Erlang(2) risk processes. Insurance Math. Econom. 22, 251--262.
  • Embrechts, P., Grandell, J. and Schmidli, H. (1993). Finite-time Lundberg inequalities in the Cox case. Scand. Actuarial J. 1993, 17--41.
  • Emery, D. J. (1973). Exit problem for a spectrally positive process. Adv. Appl. Prob. 5, 498--520.
  • Gusak, D. V. (1969). On the joint distribution of the first exit time and exit value for homogeneous processes with independent increments. Theory Prob. Appl. 14, 14--23.
  • Jacobsen, M. (2003). Martingales and the distribution of the time to ruin. Stoch. Process. Appl. 107, 29--51.
  • Jacobsen, M. (2005). Point Processes, Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston, MA.
  • Kou, S. G. and Wang, H. (2003). First passage times of a jump diffusion process. Adv. Appl. Prob. 35, 504--531.
  • Kyprianou, A. E. and Palmowski, Z. (2003). Fluctuations of spectrally negative Markov additive processes. Submitted.
  • Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line. The Dynamic Approach (Prob. Appl. (NY)). Springer, New York.
  • Paulsen, J. and Gjessing, H. K. (1997). Ruin theory with stochastic return on investments. Adv. Appl. Prob. 29, 965--985.
  • Tolver Jensen, A. (2004). The distribution of various hitting times for a shot noise process with two-sided jumps. Preprint 6, Department of Applied Mathematics and Statistics, University of Copenhagen.
  • Winkel, M. (2005). Electronic foreign-exchange markets and passage events of independent subordinators. J. Appl. Prob. 42, 138--152.