Nash equilibrium strategies of an inconsistent stochastic control problem

Abstract

In this thesis we study two research topics by using stochastic control methods in order to solve, in distinct contexts. The …rst topic presents a characterization of equilibrium in a game-theoretic description of discounting conditional stochastic linear-quadratic (LQ for short) optimal control problem, in which the controlled state process evolves according to a multidimensional linear stochastic di¤erential equation, when the noise is driven by a Poisson process and an independent Brownian motion under the e¤ect of a Markovian regime-switching. The running and the terminal costs in the objective functional are explicitly dependent on several quadratic terms of the conditional expectation of the state process as well as on a nonexponential discount function, which create the time-inconsistency of the considered model. Open-loop Nash equilibrium controls are described through some necessary and su¢ cient equilibrium conditions. A state feedback equilibrium strategy is achieved via certain di¤erential-di¤erence system of ODEs. As an application, we study an investment–consumption and equilibrium reinsurance/new business strategies for mean-variance utility for insurers when the risk aversion is a function of current wealth level. The …nancial market consists of one riskless asset and one risky asset whose price process is modeled by geometric Lévy processes and the surplus of the insurers is assumed to follow a jump-di¤usion model, where the values of parameters change according to continuous-time Markov chain. A numerical example is provided to demonstrate the e¢ cacy of theoretical results. In the second topic, we investigate the Merton portfolio management problem with non-exponential discount function and general utility function. We consider that the market coe¢ cients according to a …nite state Markov chain. The non-exponential discount in the objective function is the reason for the timeinconsistency in our topic. Since this problem is time-inconsistent we treat it by placing within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. Using a variational technical approach, we derive the necessary and su¢ cient equilibrium condition, also we provide a veri…cation theorem for an open-loop equilibrium strategies.