Minimizing ruin probabilities by reinsurance and investment: A Markovian decision approach

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.