Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:16:58.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Differentiation of some functionals of risk processes, and optimal reserve allocation

Part of: Stochastic processes Applications Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Stéphane Loisel
Stéphane Loisel*
Affiliation:
Université Lyon 1
*
∗Postal address: Ecole ISFA, 50 avenue Tony Garnier, 69366 Lyon Cedex 07, France. Email address: stephane.loisel@univ-lyon1.fr
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.

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

Keywords

Ruin theorysample path propertyoptimal reserve allocationmultidimensional risk processrisk measure

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G55: Point processes 91B30: Risk theory, insurance 91B32: Resource and cost allocation 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 379 - 392
Copyright
© Applied Probability Trust 2005 

References

Dos Reis, A. E. (1993). How long is the surplus below zero? Insurance Math. Econom. 12, 23–38.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H. U. (1988). The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance Math. Econom. 7, 193–199.Google Scholar
Gerber, H. U. (1988). Mathematical fun with ruin theory. Insurance Math. Econom. 7, 15–23.Google Scholar
Loisel, S. (2005). Finite-time ruin probabilities in the Markov-modulated multivariate compound Poisson model with common shocks, and impact of dependence. Working paper, Cahiers de recherche de l'ISFA, WP2026.Google Scholar
Picard, P. (1994). On some measures of the severity of ruin in the classical Poisson model. Insurance Math. Econom. 14, 107–115.CrossRefGoogle Scholar