Optimal Dividend Problems And Ruin Problems On Two Types Of Companies

With the improvement and development of modern financial markets, actuarial science is attracting more and more attention. In recent years, optimal dividend problems and ruin problems have become hot research topics in actuarial research for their broad application and high theoretical value. Introduction of dividend payments not only provide the theory basis for asset appraisal, but also provide theoretical support for designing and developing dividend products. Ruin problems can be used to analyze stabilities of operating performance, predict ruin probabilities and provide some warning signs of early risk.This dissertation focuses on two types of companies. One has occasional claims and constant premium rate such as insurance companies and the other has occasional gains and constant expense rate such as exploration companies. Optimal dividend problems and ruin problems are discussed for both two types of companies respectively. The research on these two problems under diffusion risk model and classical risk model has made some progress. However, further research is needed to use models which are more suitable for practice. Jump-diffusion model and its dual model proposed by Gerber et al is used to describe the surplus of two types of companies in this dissertation. With the relevant model, optimal dividend strategy, optimal dividend value function and the expected discounted penalty function at ruin are investigated. The main work are summarized as follows:Optimal dividend problem of an insurance company is studied in Chapter 2 under perturbed classical risk model with constant interest rate for an insurance company. The Hamilton-Jacobi-Bellman (HJB) equation is derived using the standard method based on the Dynamic Programming Principle. Optimal control problems are often solved under the smoothness assumption of the value function. However, proof of smoothness is not an easy work. Since the smoothness of the value function cannot be judged directly in perturbed classical risk model, emphasis is put on this problem. Firstly, the optimal value function is proved to be a viscosity solution of corresponding HJB equation. Secondly, twice continuous differentiability of optimal value function is obtained under concavity assumption on the value function and the continuity assumption on claim size distribu- tions. Finally, the closed form solution of the optimal value function is presented by the conclusion that the optimal value function is the classical solution to the HJB equation.Optimal dividend problem of an exploration company is discussed in Chapter 3 using dual jump-diffusion model in stochastic interest rate environment. Different from Chapter 2, this chapter focuses on the optimal dividend strategy. How to choose an optimal parameter of dividend barrier has been solved by Avanzi and Gerber. But, the question whether the barrier strategy is the optimal one has not got a precise and complete answer. In Chapter 3, the optimality condition under which a barrier strategy is optimal among all admissible polices is obtained. Furthermore, the optimal policy is proved to take the form of a barrier strategy in the special case that gains jumps come from a compound Poisson process with mixtures of exponential distributions.Chapter 4 investigates ruin problems of an insurance company under the perturbed compound Poisson risk model. Firstly, an integro-differential equation satisfied by the expected discounted penalty function at ruin is derived using the regeneration property of Poisson process and martingale theory. Secondly, the closed form solution of the expected discounted penalty function is derived when interest rate is constant, using the theory of Volterra integral equation. Finally, expressions of some actuarial variables are obtained by choosing different penalty functions.Chapter 5 discusses ruin problems of an exploration company under dual classical risk model with constant interest rate in the presence of a constant dividend barrier. Inspired by the expected discounted penalty function at ruin raised by Gerber and Shiu, an auxiliary function is introduced to get actuarial variables. Firstly, integro-differential equations with certain boundary conditions are derived for the auxiliary function. Secondly, explicit solutions are obtained when gains sizes are exponentially distributed. Finally, as the application of the auxiliary function, every-order moment of the time of ruin and ruin probability are derived.