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Minimizing Ruin Probabilities by Reinsurance and Investment: A Markovian Decision Approach

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Optimization, Control, and Applications of Stochastic Systems

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

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Abstract

Finite-horizon insurance models are considered where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Control problems for risk/reserve processes are commonly formulated in continuous time. In this chapter, we present a new setting which is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time Semi-Markov process (SMP) which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore, we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability. We connect this optimization problem with a controlled Markovian decision problem (MDP) over a possibly infinite number of periods. This allows one furthermore to obtain an explicit semianalytic solution for a specific case of the underlying SMP model, namely, for exponential intra-event times. It allows one also to obtain some qualitative insight into the impact that investment in the financial market may have on the ruin probability.

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Acknowledgements

Rosario Romera was supported by Spanish MEC grant SEJ2007-64500 and Comunidad de Madrid grant S2007/HUM-0413. Part of the contribution by Wolfgang Runggaldier was obtained while he was visiting professor in 2009 for the chair in Quantitative Finance and Insurance at the LMU University in Munich funded by LMU Excellent. Hospitality and financial support are gratefully acknowledged.

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Authors and Affiliations

  1. University Carlos III, Madrid, 28903, Spain

    Rosario Romera

  2. Dipartimento di Matematica Pura ed Applicata, University of Padova, Padova, 35121, Italy

    Wolfgang Runggaldier

Authors
  1. Rosario Romera
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  2. Wolfgang Runggaldier
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Correspondence to Rosario Romera .

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Editors and Affiliations

  1. , Department of Probability and Statistics, Center for Research in Mathematics, Jalisco s/n, Guanajuato, 36000, Mexico

    Daniel Hernández-Hernández

  2. , Department of Mathematics, University of Sonora, Rosales s/n, Hermosillo, 83000, Sonora, Mexico

    J. Adolfo Minjárez-Sosa

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Romera, R., Runggaldier, W. (2012). Minimizing Ruin Probabilities by Reinsurance and Investment: A Markovian Decision Approach. In: Hernández-Hernández, D., Minjárez-Sosa, J. (eds) Optimization, Control, and Applications of Stochastic Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8337-5_14

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