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Integral equations for compound distribution functions

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, 2100 Ø, Denmark.
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Abstract

The aim of the present paper is to introduce some techniques, based on the change of variable formula for processes of finite variation, for establishing (integro) differential equations for evaluating the distribution of jump processes for a fixed period of time. This is of interest in insurance mathematics for evaluating the distribution of the total amount of claims occurred over some period of time, and attention will be given to such issues. Firstly we will study some techniques when the process has independent increments, and then a more refined martingale technique is discussed. The building blocks are delivered by the theory of marked point processes and associated martingale theory. A simple numerical example is given.

Keywords

MARKED POINT PROCESSRISK PROCESSMARTINGALE REPRESENTATIONMARKOVIAN ENVIRONMENTPANJER RECURSION FORMULA

MSC classification

Primary: 60G55: Point processes
Secondary: 35L60: Nonlinear first-order hyperbolic equations
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 2 , September 1996 , pp. 388 - 399
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Asmussen, S. (1989) Risk theory in a Markovian environment. Scand. Act. J , 69–100.Google Scholar
[2] Brémaud, P. (1981) Point Processes and Queues. Springer, New York.Google Scholar
[3] Davis, M. H. A. (1984) Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models, J. R. Statist. Soc. 46, 353–388.Google Scholar
[4] Jacod, J. and Shiryayev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
[5] Møller, C. M. (1992) Martingale results in risk theory with a view to ruin probabilities. Scand. Act. J. , 123–139.Google Scholar
[6] Møller, C. M. (1993) A stochastic version of Thiele's differential equation. Scand. Act. J. , 1–16.Google Scholar
[7] Møller, C. M. (1995) Stochastic differential equations for ruin probabilities. J. Appl. Prob. 32, 74–89.CrossRefGoogle Scholar
[8] Panjer, H. H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bullet. 12, 22–26.Google Scholar
[9] Protter, P. (1990) Stochastic Integration and Differential Equations. Springer, New York.Google Scholar
[10] Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer, New York.Google Scholar
[11] Schwarz, H. R. (1989) Numerical Analysis; A Comprehensive Introduction. Wiley, New York.Google Scholar
[12] Seal, H. L. (1969) Stochastic Theory of a Risk Business. Wiley, New York.Google Scholar