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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.
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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 

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