Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T09:08:55.167Z Has data issue: false hasContentIssue false
  • English
  • Français

Ruin probabilities in a diffusion environment

Part of: Markov processes Limit theorems Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jan Grandell  and
Hanspeter Schmidli
Jan Grandell
Affiliation:
Royal Institute of Technology, Department of Mathematics, Royal Institute of Technology, SE-10044 Stockholm, Sweden
Hanspeter Schmidli
Affiliation:
University of Cologne, Department of Mathematics, University of Cologne, Weyertal 86-90, D-50931 Cologne, Germany. Email address: schmidli@math.uni-koeln.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach applied to diffusion processes, finite-time Lundberg inequalities are obtained. By change-of-measure techniques, Cramér–Lundberg approximations are derived.

Keywords

Ruin probabilityLundberg inequalityCox processdiffusion processmartingale methodchange-of-measure technique

MSC classification

Primary: 60K37: Processes in random environments 60J60: Diffusion processes
Secondary: 60G44: Martingales with continuous parameter 60F10: Large deviations
Type
Part 1. Risk Theory
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 39 - 50
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Asmussen, S., (2000). Ruin Probabilities. World Scientific, River Edge, NJ.CrossRefGoogle Scholar
[2] Björk, T. and Grandell, J., (1988). Exponential inequalities for ruin probabilities in the Cox case. Scand. Actuarial J. 1988, 77–111.CrossRefGoogle Scholar
[3] Embrechts, P., Grandell, J. and Schmidli, H., (1993). Finite-time Lundberg inequalities in the Cox case. Scand. Actuarial J. 1993, 17–41.CrossRefGoogle Scholar
[4] Gerber, H. U., (1973). Martingales in risk theory. Mitt. Verein. Schweiz. Versicherungsmath. 73, 205–216.Google Scholar
[5] Grandell, J., (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
[6] Grandell, J., (1991). Finite time ruin probabilities and martingales. Informatica 2, 3–32.Google Scholar
[7] Grigelionis, B., (1993). On Lundberg inequalities in a Markovian environment. In Stochastic Processes and Optimal Control (Friedrichroda, 1992; Stoch. Monogr. 7), Gordon and Breach, Montreux, pp. 83–94.Google Scholar
[8] Grigelionis, B., (1993). Two-sided Lundberg inequalities in a Markovian environment. Liet. Mat. Rink. 33, 30–41.Google Scholar
[9] Grigelionis, B., (1994). Conditionally exponential families and Lundberg exponents of Markov additive processes. In Probability Theory and Mathematical Statistics, eds Grigelionis, B. et al., TEV, Vilnius, pp. 337–350.CrossRefGoogle Scholar
[10] Lawrance, A. J., (1972). Some models for stationary series of univariate events. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P. A. W., Wiley-Interscience, New York, pp. 199–256.Google Scholar
[11] Palmowski, Z., (2002). Lundberg inequalities in a diffusion environment. Insurance Math. Econom. 31, 303–313.CrossRefGoogle Scholar
[12] Palmowski, Z. and Rolski, T., (2002). A technique for exponential change of measure for Markov processes. Bernoulli 8, 767–785.Google Scholar
[13] Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J., (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.CrossRefGoogle Scholar
[14] Schmidli, H., (1996). Lundberg inequalities for a Cox model with a piecewise constant intensity. J. Appl. Prob. 33, 196–210.CrossRefGoogle Scholar
[15] Schmidli, H., (1997). An extension to the renewal theorem and an application to risk theory. Ann. Appl. Prob. 7, 121–133.CrossRefGoogle Scholar