Optimal stochastic control and risk minimisation in Insurance

Abstract

The thesis examines a generalised problem of optimal control of a firm through reinsurance, dividend policy and convex risk minimisation in the presence of market friction. The major mathematical tool applied is the theory of stochastic control for jump-diffusions. In the absence of intervention the financial reserves of the firm are assumed to evolve according to a stochastic differential equation with a jump component. In the second and third chapters, the objective is to derive reinsurance and dividend policies that maximize the expected total discounted value of a spectrally negative process in incomplete markets. The assumption is that transaction costs are incurred whenever dividends are paid out. Several verification theorems are derived and proved for combined singular and impulse control. The verification theorems are new results which provide a federative approach to the analysis of control problems involving transaction costs in finance and insurance. Two methodologies are examined for risk minimisation. First, we investigate risk minimisation using zero-sum stochastic differential game theory in the presence of transaction costs. Our major contribution in this direction is that we have investigated, for the first time in the literature, a singular control problem for jump 7 diffusion stochastic differential games. Hamilton-Jacobi-Bellman-Isaacs variational inequalities (HJBIVI) are formulated and proved for the case of zero-sum stochastic differential games. The notion of HJBIVI is later on extended to the more general case of Nash equilibrium. Minimisation of risk is also studied using g-expectation. In this case a five step scheme is formulated. The scheme constitutes a mechanism for solving forward-backward stochastic differential equations. The solution provided by such a scheme minimises risk of terminal wealth of an insurance company. An existence and uniqueness theorem for the solution is provided. Several examples are discussed, throughout the thesis, to illustrate the theory.