Studies On Optimal Control Problems For Insurance Risk Models

The problem of optimal dividends distribution of insurance company has recently been gaining a lot of attention in actuarial science. From a certain point of view, the value of discounted dividend payment is equivalent to the value of the company. Meanwhile, studies of dividends distribution provide theoretical support for designing dividend products. This thesis is focused on the issue of optimal dividends distribution. Under various factors and constraints, we study the optimal strategies for insurance company with controllable reinsurance, investment and equity issuing. The thesis is organized as follows:In Chapter 1, we introduce the basic risk models, concepts and related stochastic control theory.In Chapter 2, we investigate the diffusion models for insurance company with con-trollable reinsurance, investment and dividend distribution. We use the expected dis-counted value of dividend payment plus the terminal value as the criterion for optimality. Meanwhile, we allow the company taking an extra insurance business from other com-pany. Terminal value of the company at the time of the bankruptcy could be the value of non-liquid assets. We apply the theory of dynamic programming to solve the problem and find the optimal policies.In Chapter 3, we study a model for a insurance company with constraint on risk control. The objective of the insurer is to find a business policy and a dividend payment scheme so as to maximize the expected discounted value of dividend payment, and the expected present value of an amount which the insurer earns until the time of ruin. The object function can be used to find the balance between optimizing the value of dividend and maximizing the ruin time. First, we study a model for a insurance company with constraint on risk control. Then, under the consideration of both internal competition and the time value of ruin, we obtain a non-linear controlled diffusion model. By solving the constrained HJB equation, we obtain the explicit expression for value function and the corresponding optimal strategies.In Chapter 4, we study the problem of optimal proportional reinsurance and divi-dend payment with transaction costs and internal competition. Due to the presence of a fixed transaction cost, the resulting problem becomes a stochastic impulse control prob-lem. By solving the corresponding quasi-variational inequality, we solve the problem explicitly.In Chapter 5, we consider the optimal control problem of the insurance company with excess-of-loss reinsurance policy. The management of the company controls the reinsurance, dividends payout as well as the equity issuance processes to maximize the expected present value of the dividends minus the expected present value of the equity issuance until the time of bankruptcy, we firstly construct two categories of suboptimal models. one is the classical model without equity issuance, the other never goes bankrupt by equity issuance. Then we identify the value functions and the optimal strategies with the corresponding solution to the suboptimal models depending on the relationships between the coefficients.