Ruin Probabilities For An Insurer Investing Its Money In The Stock Market

The classical risk model has a lot of restriction with the reality of the insurance company developing with the financial market. In fact, the profit of investment in the insurance company decides the final profit of the company. So a risk model with investment has become a hot topic in the present actuarial science and mathematical research. In this paper, we study the ruin probability in the classical Cramér-Lundberg model, where the company invests a constant fraction of its money in a stock, which is described by geometric Brownian motion. We prove a certain integro-differential equation of the survival probability by use of the It? formula and prove the existence of the solution of the equation. And then we give the Laplace equation of the survival probability to get the numerical solution through the computer when the distribution of the claim size has been given. Furthermore, we also study a risk model with investment, whose proportion investing in a stock is not a constant number, but a function of the time t . We will give an integro-differential equation of the function of the time and the surplus process in the latter model. We introduce Sobolev space, in which we can get the existence and uniqueness of the weak solution of the integro-differential equation.