Research On Stochastic Control Problem Of Several Classes Of Risk Model

In this Ph.D. thesis, we investigate the dividend payments and capital injections problem of several classes of risk models which include compound Poisson risk model, compound Poisson risk model perturbed by diffusion, two dimensional compound Poisson risk model and more general risk model-piecewise deterministic Poisson risk model. It is composed of nine chapters.Chapter1is dedicated to the overview of the historical background, the present development of problems concerned with dividend payments and capital injections of several risk models. Meanwhile, the main results and innovative contributions of this thesis are introduced.In Chapter2, we analyze the optimal dividend payment and capital injection problem for the classical risk model perturbed by diffusion. The objective of the corporation is to maximize the discounted dividend payments minus the penalized discounted capital injections until the time of ruin. The problem is formulated as a stochastic control problem. By solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation, we obtain the optimal dividend and capital injection strategy of the problem. Under some reasonable assumptions, the optimal dividend strategy prescribes to company using a barrier strategy and pay no dividends when the reserve is below some critical level b and to pay out everything that exceeds b. The optimal injection strategy is described by the optimal lower and upper injection barrier. We solve this problem explicitly in the case of exponential claim amount distributions.In Chapter3, we discuss the impulse stochastic control optimal dividend problem for the classical risk model perturbed by diffusion. With each dividend payment there is a proportional cost and a fixed cost. The objective of the corporation is to maximize the cumulative expected discounted dividends payout until the time of ruin. It controls the timing and the amount of dividends paid out to the shareholders. By solving the corresponding quasi-variational inequalities, we obtain the optimal dividend strategy. Furthermore, we solve this problem explicitly in the case of exponential claim amount distributions of this jump-diffusion model. It turns out that there can be essentially three different solutions depending on the model’s parameters and the costs. Specifically,1. Whenever the reserve reaches an appropriate level barrier**, it is reduced to x>0through a dividend payment, and the process continues.2. Whenever assets reach an appropriate level x’, everything is paid out as dividends immediately, and the ruin occurs.Chapter4extends the known result due to Belhaj who find the optimal dividend policy is of a barrier type for a jump-diffusion model with exponentially distributed jumps. It turns out that there can be essentially two different solutions depending on the model’s parameters.1, the initial capital should be paid out as dividend immediately, and the ruin occurs.2, the optimal strategy is to pay everything in excess of an appropriate level b, whenever the reserve reaches the level, and the process continues. It also deals with the optimal control problem for the jump diffusion process with solvency constraints. The objective of the corporation is to maximize the cumulative expected discounted dividends payout with solvency constraints. It is well known that under some reasonable assumptions, optimal dividend strategy is a barrier strategy, i.e., there is a level b’so that whenever surplus goes aboveb*, the excess is paid out as dividends. However, the optimal level b*may be unacceptably low from a solvency point of view. Therefore, some constraints should be imposed on an insurance company such as to pay out dividends unless the surplus has reached a level b0>b*. We show that in this case a barrier strategy atb0is optimal.In Chapter5, we deal with the optimal dividend problem for a two-dimensional compound Poisson risk model which constructs correlation among the two claims. We extend the two-dimensional risk model studied in the recent literature. The objective of the corporation is to maximize the expected discounted dividend payout until the time of ruin. By solving the corresponding HJB equation, we obtain the optimal dividend strategy of the problem. We solve this problem explicitly in the case of exponential claim amount distributions in two examples. In Chapter6, we study the optimal dividend payment and capital injection problem for a two-dimensional compound Poisson risk model which constructs correlation among the two claims. The objective of the corporation is to maximize the discounted dividend payments minus the penalized discounted capital injections. By solving the corresponding HJB equation, we obtain the optimal dividend strategy of the problem. We solve this problem explicitly in the case of exponential claim amount distributions in the example.In Chapter7, we investigate impulsive stochastic control for the optimization of dividend payment and capital injection of the classical risk model. The objective of the corporation is to maximize the expected discounted dividends payout minus the equity issuance until the time of bankruptcy. It turns out that the control problem is associated with qualitatively different optimal capital injection strategies, depending on the problem’s data. One allows for no capital injection and the other allows for capital injection. We solve this problem explicitly in the case of exponential claim amount distributions. It is shown that there can be essentially seven different solutions depending on the model’s parameters and the costs.Chapter8discusses the classical risk model with impulsive dividend strategy. An integro-differential equations for the Gerber-Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the Laplace transform of the time of ruin, the distribution of the surplus immediately before ruin and the deficit at ruin. Moreover, the distribution of the number of dividend is presented.Chapter9deals with optimal dividend payment problem in the general setup of a piecewise-deterministic compound Poisson risk model. The objective of the insurance business model under consideration is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in special cases of risk models in the literature. We prove with the generality of the piecewise-deterministic compound Poisson process the following results. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. The case of unrestricted payment scheme gives rise to a singular stochastic control problem. We solve the associated integro-differential quasi-variational inequality, which produces an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we offer general solutions to both dividend policies. Explicit expressions as well as numerical examples are presented for a number of practical applications.