The Research On Ruin Probability Of The Risk Model With Random Premium Rate

In this thesis,introducing the stochastic premium rate, we consider several new models based on the classical risk model. Risk reserve model on condition that premium rate is an alternating renewal process, research of survival probability in Markov-modulated Risk model with alternately premium rate,and research on ruin-probability of insurance based on queuing theory. We mainly study the adjustment coefficient, the ultimate ruin probability and survival probability, the relationship between the ruin probability and premium rate, the relationship between the limit distributions of waiting time and ruin function.Frist,under the condition of continuous time,we consider the risk model which the premium rate is an alternating renewal process and the occurrence of claim is a homogeneous Poisson process.Analyzing the proposed model, we obtain the stationary increment properties of profit process and the statistical character of risk process;by martingale approach, the formula of the ruin probability and the Lundberg inequality are derived; in accordance with the negatively correlated relation between expected profit and ruin probability, the smallest upper boundary of ruin probability under optimal threshold condition is obtained, and by the concrete example, the relationship between the ruin probability, the initial capital, premium rate and time threshold is discussed.Secondly, we consider a Markov-modulated risk model with alternately changing premium rate, when premium rate is controlled by Markov process with the two-state, the integral-differential equations on survival probabilities are obtained; the equations of Laplace transforms of survival probability is discussed,and according to the non-negative quality of the root of the characteristic equation on coefficient matrix, explicit formulae for survival probabilities are given when the initial reserve is zero; when the claims are exponentially distributed, we obtain the explicit expressions of survival probabilities.Lastly, we research ruin probability of insurance based on the Sparre Anderson risk model.Considering ruin probability of the risk model and introducing a queuing model, we derive the equivalent relationship between the limit distributions of waiting time and ruin function;When the loss process of insurance company is treated as a random walk process, we obtain the upper bound of ruin probability in terms of some techniques from martingale theory;according to the feature of the risk model, the recursive and integral equation for the ruin probability is obtained.In particular, under the mathematical induction, we get an upper bound of ruin probability which is smaller than that in the Lundberg inequality, and numerical example with exponential claim is given.