Risk Theory With Variable Premium Rate Under Interest Force

In Actuarial Mathematics, the study of ruin theory has been the centre of interest of risk theory. In 1903, Lundberg proposed an important stochastic process, namely, Poisson process in his doctor paper. On the base of his, Cramer developed strict stochastic process theory. After Cramer, Feller and Gerber introduced Renew and Martingale methods. All of these are important methods in the study of ruin theory.In Lundberg-Cramer model, insurer company receives the premium with a constant rate, the occurrence of claim is a Poisson process. The risk models are extensively discussed by many authors. Gerber considered uncertainly profit and presented a ruin model perturbed by diffusion. Many papers improved it from the arrival of the guarantee and the occurrence of claims. Other fellows considered all kinds of insurance and proposed multiple risk models.In this thesis, we improve the assumption that the insurance company receives the premium with a differentiable rate. At the same time, we instead Cox process of Poisson process to describe the occurrence of claims and consider the diffusion. It should be noted that the company also receives interest of its reserves with a constant force. Think of all of these, in the considered Cox risk model, we discuss equations for the ultimate survival probability as well as approximations and upper and lower bound.Based on it, the probability of ruin is investigated under the influence of a variable premium rate and the stochastic rates of interest. In the present risk model, the occurrence of claims is described by the Cox process. When the interest process is a Brownian motion with drift, we derive an integro-differential equation for ruin probability. Applications of the integro-differential equation are given.