On Some Ruin Problems For Risk Models With Stochastic Return On Investment

Risk theory plays an important role in actuary. It focuses on the study of all kinds of risk models related to insurance affairs, and thus is of all importance for modeling risk in the study of risk theory. Originated from Lundberg [59], risk theory has received intensive attraction in a large amount of literature. One aspect of aforementioned literature locates in the generalizations of risk model, some new risk models are introduced, including renewal risk model, perturbed classical risk model, discrete time risk model, risk model with compounded asset and the cross-research between the actuary and finance. Some others focus on the applications of stochastic control theory on the risk study in different ways, such as the investment, reinsurance, dividend payment, new insurance policy et al. See Hipp [42] for a complete and detailed survey.In my thesis, some ruin problems for risk models with stochastic return on investment and some optimal control problems related to ruin are discussed. We consider in this thesis two main cases: the discrete time case and the continuous time case. The whole thesis is divided into three parts (seven chapters). The first chapter introduces some essential background and main results.In the first part (Chapter 2 and Chapter 3) we study the continuous risk model with stochastic return on investments. Our discussion consists of two risk models: the risk model with stochastic premium and the jump-diffusion risk model. Many existing works assume that the premium income process to be a linear function of time t. However, in Chapter 2, the premium income process is assumed to be a stochastic process (compound Poisson process) and the model in this chapter can be regard as a generalization of the one in Paulsen[68]. Moreover, the price process of risk asset is assumed to be an exponential Levy process. By studying the skeleton of underlying risk model, we reduced the compounded risk model into a discrete time risk model, the explicit expression for ruin probability and the bounds for ruin probability are derived. Differential-integral equations for the ruin probability are also investigated in some special cases and some examples are presented to illustrate the applications of these equations. In Chapter 3, we consider the perturbed risk process with stochastic premium income in two investment ways: invest on stock market and invest on bond market. When the surplus process is invested on the stock market, by martingale approach and the theory of stochastic analysis, an integral-differential equation for the Laplace transform of ruin time is obtained and some asymptotic behavior of ruin probability are derived in the exponential claim case. While the surplus process is invested on the bond market, we use the Vasecik model for modeling the interest rate of bond market. Besides, we also study the decomposition of ruin probability. The Vasecik model is widely used in financial mathematics for modeling the term structure of stochastic interest rate when it comes to the pricing of the bond market. However, it is this paper that consider the problems under such interest rate structure.In the second part (Chapter 4 and Chapter 5) we investigate some optimal problems for the risk model that has the possibility to invest on the risky market. The first problem is the optimal investment and reinsurance strategy for maximizing the expected terminal utility of an insurer. We assume the surplus process of insurer is a perturbed risk model with stochastic premium income, and it can be invested in a risky market. By solving Hamilton-Jacobi-Bellman equations related to our optimal problems, the closed form expressions for the optimal strategies and the value functions are derived. We also studied the asymptotic behavior of the optimal strategy for an insurer when the initial surplus tends to infinity. By minimizing the upper bound for ruin probability we obtained an maximized adjusted coefficient and an corresponding constant strategy. We also show that the constant strategy is asymptotically optimal, which indicates that the optimal investment strategy for an insurer to minimize the ruin probability is very conservative. Therefore, the criterion for minimizing the ruin probability is not so appropriate when it comes to the optimal investment.In the third part (Chapter 6 and Chapter 7), ruin problems for the discrete time risk model are investigated. The claims are assumed to be independent in the classical risk model and some of its generalizations. However, with the developing of insurance affairs, it is necessary to study the correlated claims case. In Chapter 6, a discrete time risk model with stochastic interest rate is introduced. We assume that the net losses of an insurer follow the AR(1) structure and the interest rate follows a time homogeneous Markov chain with infinite phase space. The main difference between our model and the one in [14] is that the net losses in our model are assumed to be depended and the main difference between our model and the one in [85] is that we replace the assumption that the interest rate is constant with a Markov chain with infinite phase space. By martingale approach and recursive method, upper bounds for ruin probability are derived. In Chapter 7, we investigate the optimal dividend payment for an discrete time risk model with stochastic interest rate. By dynamic approach and Bellman's optimal principle, we obtained Bellman's equation for our optimal problem, and finally we give the algorithm for our problem by solving a practical example.