Research On Ruin Probability, Optimal Portfolio And Reinsurance Problems For Several Classes Risk Models

The insurance company plays an important role in the financial system. The insurance company helps the insurers to reduce their risks. Meanwhile, the insurance company earns the profit and increases the efficiency of the fi-nancial system. This paper investigate the ruin probability and the optimal portfolio and reinsurance problems for several classes risk models by stochastic control theory, stochastic process and martingale theory. The main ideas and contributions of this thesis is organized as follows.1. In chapter 2, we consider a Markov-modulated Pascal model with ran-domized decisions on paying dividends under stochastic interest rates. Finite and infinite time ruin probabilities and the joint distribution of the ruin time, the surplus before ruin and the deficit at ruin in this model are studied, and the recursive formulas and the integral equations for them are given, respectively. When the insurer will pay a unit dividend with probability 1 if the surplus is no less than a non-negative integer b, generalized Lundberg inequalities for the infinite time ruin probability are derived by inductive approach.2. In chapter 3, we obtain integral equations and an expression of the Gerber-Shiu discounted penalty function in the delayed renewal risk model with constant interest in terms of the corresponding Gerber-Shiu function in the or-dinary renewal risk model with constant interest. Finally, the exact expressions for the ruin probability in a special case of the delayed renewal risk processes with constant interest is derived.3. In chapter 4, considers the compound Poisson risk model with a threshold dividend strategy and proportional investment. The goal here is to investigate the expected discounted dividend payments and the expected penalty-reward function. Integro-differential equations with certain boundary conditions are derived. As closed-form solutions do not exist, a numerical sinc method is proposed. Finally, some examples illustrating the procedure are presented.4. In chapter 5, considers the expected discounted penalty function for the compound Poisson risk model with time-changing. Integro-differential equation is derived. The Elzaki transform of the expected discounted penalty function is obtained from the integro-differential equation. Finally, a numerical example is provided to illustrate the impact of the time-changing on the ruin probability.5. In chapter 6, we investigate a robust optimal portfolio and reinsurance problem under inflation risk for an ambiguity-averse insurer (AAI), who worries about uncertainty in model parameters. We assume that the AAI is allowed to purchase proportional reinsurance and invest his/her wealth in a financial market which consists of a risk-free asset and a risky asset. The objective of the AAI is to maximize the minimal expected power utility of terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategies are obtained.