The Study Of Penalty Function Of Insurance Complany Under The Stochastic Rate

Considering with the opening of financial market, interest rate’s marketization has become the main content of the reform of the financial markets, interest rates will grad-ually to be control by the market. So under the background of interest rates randomized, we promote the classic model of risk management in this paper, and we mainly stud-ied the probability density function of penalty function and the value of the expected of the discounted penalty function function under the various random interest rates by the method of martingales, and compare the results with the predecessors. Finally, we also has carried on the further promotion for the claim, consider the claims process to be a Markov jump processes and we get the stochastic difference equations of the value of the expected of the discount penalty function at any time before the Bankruptcy moment, and also gives the stochastic difference equations of some special situations. The main contents are arranged as following:The main contents are arranged as following:In the first chapter, we mainly introduce the social background of risk management and the realistic problem what we faced now, then we also described the development of the subject briefly and the researching status of domestic and foreign.The second chapter, first of all introduces some basic knowledge of the modern the-ory of probability and stochastic calculus, which includes:probability and measure, ran-dom variables, stochastic processes, martingale, Ito’s formula, etc. The second part intro-duces one of the most basic stochastic process in the risk management-Poisson process. The third part is the review of the mathematical definition of classical risk model.The third chapter, reconsider the density function of the discount penalty at bankrupt-cy time and the expectations of the discount penalty at bankruptcy time from the aspect of the martingale, and comparing with previous related literature.The fourth chapter, we further promote the process of interest rate, considering when the logarithmic of the interest rates is a Brownian motion with drift plus a compound Pois-son process at first, we also get the density function of the discount penalty at bankruptcy time and the expectations of the discount penalty at bankruptcy time; Second, when in-terest rates process is exponent levy process, we get the density function of the discount penalty at bankruptcy time and the expectations of the discount penalty at bankruptcy time again.The fifth chapter, we further promote the process of compensation, consider total claims process as a Markov jump process, get the stochastic differential equations of the surplus function, and stochastic difference equations of the expection of the discounted plenty function at any time before bankruptcy, and calculation the SDE when penalty function is 1 constantly with initial funding to be 0.