Stochastic optimal control and stochastic differential games: applications in insurance

Abstract

The present thesis is divided into two parts, The first part begins with the development of a new approach to study the problem of optimal investment under asymmetric information. This approach heavily relies on stochastic optimal control techniques and in particular on the use of the Hamilton-Jacobi-Bellman equation. Then, we turn our attention to the introduction of inside information aspects to the insurance/reinsurance market. This accomplished by considering two firms: an insurer and a reinsurer and letting one of the firms, the insurer, posses some additional information which is hidden from the reinsurer. By employing the aforementioned approach, we are able to treat the problem of maximizing the expected utility from terminal wealth, for both firms, by taking explicitly into account their different information on the optimal decisions of the insurer. The aim of the second part is to study a robust-entropic optimal control problem between an insurance firm and Nature. However, a major obstacle arises, as the value of this problem is associated with a fully nonlinear partial differential equation that may not admit smooth solutions. In order to overcome this difficulty, we write this general problem as a normal form zero sum stochastic differential game with two players and resort to the classical theory developed by Fleming and Souganidis[42] in order to prove that the associated Bellman-Isaacs partial differential equation admits a unique continuous viscosity solution, which is the Nash value of the game. Furthermore, we state and prove a general verification theorem that allows to characterize the optimal controls of the players. Finally, we provide the connection of the robust-entropic control problem with the theory of convex risk measures and we conclude with the study of the asymptotic behavior of the aforementioned Bellman-Isaacs equation.