Several Risk Models In Finance And Insurance

The paper investigate the ruin problem of several risk processes in finance and insurance by the renewal argument, Markov process, stochastic control and martingale theory. The upper bound of ruin probability, the distributions of the surplus immediately before ruin and the deficit at ruin, the property of the expected discounted penalty function at ruin and the optimal proportion of new risk business are discussed. The main ideas and contributions of this thesis are as follows:1. Extending the deterministic premiums income in the classical risk model to the compound Poisson process, and describing the disturbance of the stochastic factors by the Wiener process, we consider the double compound Poisson risk model perturbed by diffusion. Making use of the character of the stationary and independent increments of the risk process, we get the general formula and the Lundberg inequality of the ruin probability, and analyze the effect of the initial surplus, the income of premium and the claim payment on the ruin probability and the adjustment coefficient by a numerical example. The integral equations of the distributions of the supremum surplus before ruin and the deficit at ruin are given by the homogeneous strong Markov property of the risk process.2. The expected discounted penalty function in Erlang(2) risk process is discussed. Introducing the constant interest rate into the Erlang(2) risk model, we derive the integro-differential equaton for the expected value of the penalty, and the recursive formula for the joint distribution of the surplus immediately before ruin and the deficit at ruin by the renewal argument. And the defective renewal equation, the series representation of the penalty function in the special interest-free case are given. Next, the Erlang(2) risk process with a new threshold dividend strategy are developed. Using certain mathematical technique, we get and solve the integro-differential equation satisfied by the discounted penalty function. The relation between the discounted penalty functions with or without the threshold dividend is obtained. 3. The Cox risk models in a markovian environment are investigated, namely, the intensity process of the Cox process is markovian jump process. Firstly, in the Cox risk model where the constant interest force is included, we obtain the integral equation for the conditional expected value of the penalty and the expected value of the penalty which is in the stationary case by the backward differential argument. Secondly, we consider the penalty function of the Cox risk model with the premium rate which varies with the claim intensity and disturbed by diffusion. The defective renewal equation and asymptotic property for the expected value of the penalty are given in certain circumstances. Finally, we establish the Cox risk process with two kinds of risk business. Further, there is a certain correlation between the claim arrival processes of the different risk business. The estimation of sharp upper bound of the ruin probability are deduced by the martingale technique.4. We consider the discounted penalty function in a kind of the risk model with time-correlated claims. In this model, the main claim can induce the by claim which may happen with the main claim simultaneously with probabilityθ, and may delay to next time epoch with probability 1—θ. The integro-differential equation of the discounted penalty function is solved by the Rouché's theorem and Laplace transform, and the numerical result is given.5. The interest force is introduced into a kind of control problem of the insurance risk. The optimal proportion of new risk business under constant interest rate is studied. Optimal is meant in the sense of minimizing the ruin probability of the insurance company. The risk business are assumed to follow a Brownian motion with drift. We derive and solve the Hamilton-Jacobi-Bellman equation by the stochastic control method. The close-form of the minimum ruin probability and the optimal proportion are found. Finally, the influence of the interest rate and the initial capital on the probability of ruin and the optimal proportion is discussed by the numerical example.