Optimal control problems and their applications to insurance

In this thesis, we study a few optimization problems for an insurance company whose purpose is to maximize profits and/or minimize risks. Our results are collected in three chapters. In the first chapter, we analyze the optimal investment and reinsurance problem of a company, endowed with forward dynamic utilities, in a stochastic factor model that allows for a double dependence between the financial and insurance markets. Precisely, we assume that the financial asset price and the insurance losses are both affected by a common stochastic factor which is described by a continuous time finite state Markov chain or a diffusion process. We construct a family of forward dynamic exponential utilities and we characterize the optimal portfolio strategy and the optimal proportional level of reinsurance. We perform some numerical experiments to further investigate our results. Moreover, we compare the forward approach with the classical one based on backward utilities, both analytically and numerically. We also discuss an extension of the conditional certainty equivalent. In the second chapter, we study the dividend maximization problem and the ruin minimization problem, under the constraint that the terminal surplus of the insurance company follows a normal distribution with a given mean and a given variance, which may be set, e.g., to realize a Value at Risk or Expected Shortfall at some pre-specified confidence level. We suppose that the surplus is modeled by a Brownian motion with drift. When the company is allowed to distribuite dividends, we seek to maximize the expected discounted dividend payments or to minimize the ruin probability under the terminal distribution constraint. We find explicit expressions for the optimal strategies in both cases, when the dividend strategy is updated at discrete points in time and continuously in time. Instead, if the company buys reinsurance for part of its claim, we investigate the reinsurance retention level that minimizes the ruin probability and allows the net collective to achieve the target distribution. Due to the fact that updating reinsurance contract is a complicated matter from the practical point of view, we study the case where the reinsurance retention level can be modified only once over a fixed time interval, typically of one or two years. In this setting, we find out that an admissible strategy is chosen at time zero, and we explicitly characterize the ruin minimizing strategy. We also discuss the implications of mantaining the initial retention level over the whole period and give the idea of how to deal with several strategy updates. In the third chapter, we discuss the indifference pricing problem of a pure endowment (namely a contract that yields a fixed amount at maturity, provided the policyholder is alive at that time) for an insurance company, whose preferences are described by an exponential utility function. We propose a modeling framework where the mortality intensity of a reference population is stochastic and the risky asset price evolves according to a jump diffusion affected by regime changes. We determine the optimal investment strategies, with and without the insurance policy, and characterize the indifference price as a classical solution to a linear PDE with a suitable nal condition and in terms of its probabilistic representation via an extension of the Feynman-Kac formula. Furthermore, we also investigate the indifference price for a portfolio of pure endowments and for a term life insurance. Finally, some numerical experiments are performed to address sensitivity analyses.