Research On Optimal Decision For Insurers Based On Time-inconsistency And Constraint

Stochastic optimal control or dynamic programming method is a powerful tool to solve many dynamic optimization problems in the economics and finance. With the economic and financial theory are unceasingly rich and the development, a lot of time-inconsistent stochastic control problems appears. The time-inconsistency means that the Bellman Optimality Principle does not hold, as a consequence, dynamic programming cannot be applied. Therefore, it is very necessary to s-tudy the time-inconsistent stochastic control problems, especially, to study their time-consistent control or strategy. The celebrated dynamic mean-variance model and optimal investment and consumption under hyperbolic discounting are both time-inconsistent stochastic control problems. In addition, many of the economic and financial models need to consider some realistic restrictions. As a result, the stochastic optimization problems with constraints are also very important and are challenging research fields. This thesis considers two time-inconsistent stochastic control problems and a stochastic optimization problem with the constraint. First, we study the time-consistent investment-reinsurance strategy. Second, we consider the time-consistent dividend-payment strategy. Finally, we discuss the dividend optimization problem with solvency constraint.In Chapter1, we firstly introduce the background of the time-inconsistent stochastic control problems and stochastic optimization problems with constraints. Then state the main results of this thesis. Finally, we list some preliminaries to solve the time-inconsistent stochastic control problems.Chapter2is devoted to studying the time-consistent investment and reinsur-ance strategies with state dependent risk aversion. It is assumed that the surplus process is approximated by a diffusion process. The insurer can purchase propor-tional reinsurance and invest in a financial market which consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. Under these, we consider two optimization problems, an investment-reinsurance problem and an investment-only problem. In particular, when the risk aversion depends dynamically on current wealth, the model is more realistic. Using the approach developed by Bjork and Murgoci (2010), the time-consistent strategies for the two problems are derived by means of corresponding extension of the Hamilton-Jacobi-Bellman equation. The time-consistent strategies are de-pendent on current wealth, this case thus is more reasonable than the one with constant risk aversion.In Chapter3, we study the time-consistent dividend strategy with non-exponential discounting in a dual model. We consider a dividend-payment problem for a com-pany with non-exponential discounting, whose surplus process is described by a dual model. The target is to find a dividend strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the ruin time of the company. The non-exponential discount function results in the problem of being time-inconsistent. But we seek only the time-consistent strategy, which is an equilibrium strategy derived by taking our problem as a non-cooperate game. Extended Hamilton-Jacobi-Bellman equation system and verification theorem are provided to derive the equilibrium strategy and the equilibrium value function. For the case of pseudo-exponential discount functions, closed-form expressions for the equilibrium strategy and the equilibrium value function are derived. In addition, some numerical illustrations of our results are showed.In Chapter4, we investigate the dividend optimization for jump-diffusion with solvency constraint. Assuming that the surplus of insurer follow a jump diffusion model, we consider a dividend optimization problem under the solven-cy constraints or ruin probability constraint. Prom, the existing literature, if not considering bankruptcy probability constraint, the optimal dividend policy under jump diffusion model is a barrier strategy. When the bankruptcy probability con-straint is considered, the dividend optimization problem is difficult to solve. We use stochastic analysis and partial differential-integral equation to consider a div-idend optimization problem with constraint, by analyzing some properties of the ruin probability, we present the optimal strategy and optimal value function of the dividend optimization problem.