Study On The Ruin Probability And Optimal Investment Of Insurance Company Under Integrated Risk Model

It is well known that the success of an insurance company depends not only on its insurance business, but also on how well the company invests its reserve. The risk such a company faces arises both from ponential losses on the financial market and from unexpectedly high insurance claims. Consequently, an integrated risk model incorporating the dynamics of the financial markets and the insurance portfolio as well as the interaction between them is needed. Ruin probability is used in this paper as a risk measure of an insurance company. We consider a stochastic model for the wealth of an insurance company which has the possibility to invest into a market, and the total insurance claim amount is modeled by a compound Poisson process.Firstly, we add the interest rate into the classical Lundberg-Cramér model. The discounted value of outstanding claims reserve is studied by the queuing model with constant interest. The distribution function and characteristic function of the discounted value of outstanding claims reserve in a given period of time are obtained. Furthermore, we give an upper bound of distribution function, the moments of the discounted value of outstanding claims reserve are given as well. We further investigate the infinite time ruin probability of the Cramer-Lundberg model with constant interest force. Exponential type upper bounds for the ruin probability are derived by martingale techniques. Classical Lundberg-Cramér model is generalised. Under the condition that the interest rate is uncertain, which is described by stochastic process, the surplus of insurance company is modeled by classical renewal risk model. Provided that the claims have regular varying-tailed distributions, the tail equivalence is obtained by the method of randomly weighted sums. The method is different from traditional ways. An effective approach to estimate the ruin probability is obtained and the classical results are generalized.Secondly, under the condition that all the assets are invested in stock and risk-free bond markets respectively, this paper investigates the infinite time ruin probability and portfolio investment of insurance company. Provided that the discounted value of risky investments is a constant family, an exponential type upper bound for the ultimate ruin probability can be obtained by the martingale approach, the optimal portfolio investment strategy can be derived as well. The aim of this paper is to provide a rational strategy which can control the risk of insurers. An example is employed to illustrate the results. Provided that two subsidiary companies of an insurance company are allowed to invest certain amount of money in some risky market and the value of risky investments is a constant family. An exponential type upper bound for the ultimate ruin probability can be obtained by the martingale approach, which can control the risk of insurance company. The optimal portfolio investment strategy can be derived, which can minimize the upper bound of ruin probability.Thirdly, we investigates the infinite time ruin probability under the condition that the company is allowed to invest certain amount of money in some stock market, and the remaining reserve in the bond with constant interest force. The total insurance claim amount is modeled by a compound Poisson process and the price of the risky asset follows a general exponential Lévy process. Exponential type upper bounds for the ultimate ruin probability are derived when the investment is a fixed constant, which can be calculated explicitly. This constant investment strategy yields the optimal asymptotic decay of the ruin probability. Some examples are applied to illustrate the results. We provide an approximation of the optimal investment strategy, which maximizes the expected wealth of the insurance company under a risk constraint on the Value-at-Risk.Lastly, we investigate the infinite time ruin probability under the condition that the company is allowed to invest certain amount of money in some stock market and the remaining reserve in the bond with constant interest force. Through the properties of Brownian motion and discrete embedded method, the integral equations for ruin probability are derived under the assumptions that the stock price follows geometric Brownian motion or Lévy process. The method for explicitly computing the ruin probability and penalty function is obtained.