Minimizing Ruin Probabilities by Reinsurance and Investment: A Markovian Decision Approach
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.
KeywordsFinancial Market Asset Price Recursive Relation Risk Process Premium Rate
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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