Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T12:17:46.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic differential equations for ruin probabilities

Part of: Hyperbolic equations and systems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Christian Max Møller
Christian Max Møller*
Affiliation:
University of Copenhagen
*
∗Postal address: Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparken 5, Copenhagen 2100 Ø, Denmark.
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.

Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

Keywords

POINT PROCESSMARTINGALE REPRESENTATIONMARKOV PROCESSMARKOVIAN ENVIRONMENT

MSC classification

Primary: 60G55: Point processes
Secondary: 35L60: Nonlinear first-order hyperbolic equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 74 - 89
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (1984) Approximations for the probability of ruin within finite time. Scand. Actuarial J., 31–57.Google Scholar
Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Actuarial J., 69–100.Google Scholar
Asmussen, S., Petersen, S. S. (1988) Ruin probabilities expressed in terms of storage processes. Adv. Appl. Prob. 20, 913–916.CrossRefGoogle Scholar
Brémaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Chung, K. L. and Williams, R. J. (1990) Introduction to Stochastic Integration, 2nd edn. Birkhäuser, Basel.Google Scholar
Cramer, H. (1955) Collective risk theory. Skandia Jubilee Volume, Stockholm.Google Scholar
Dassios, A. and Embrechts, P. (1989) Martingales and insurance risk. Commun. Statist .-Stoch. Models 5, 181–217.Google Scholar
Davis, M. H. A. (1984) Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B46, 353–388.Google Scholar
Delbaen, F. and Haezendonck, J. (1987) Classical risk theory in an economic environment. Insurance: Math. Econ. 6, 85–116.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Grandell, J. (1990) Aspects of Risk Theory. Springer-Verlag, New York.Google Scholar
Petersen, S. S. (1989) Calculation of ruin probabilities when the premium depends on the current reserve. Scand. Actuarial J., 147–159.CrossRefGoogle Scholar
Reinhard, J. M. (1984) On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. Astin Bulletin 14(1), 23–43.Google Scholar
Rogers, L. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Vol. 2: Ito calculus. Wiley, New York.Google Scholar
Seal, H. L. (1974) The numerical calculation of U(w, t), the probability of non-ruin in an interval (0, t). Scand. Actuarial. J., 121–139.Google Scholar
Smith, G. D. (1985) Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press.Google Scholar