Heavy-tailed Distribution Under The Investment Risk Model

We consider risk model, where the company is allowed to invest their money in risk market. The market contains both riskless investment (e.g. bond)) and risk investment (e.g. stock) suppose that the riskless investment rate is a constant r ,and the risk investment is described by geometric Brownian motion, the ruin probability is minimized by the choice of suitable investment strategy. We start from an integro-differential equation (Bellman equation) for the maximal survival probability. First, we proved the existence of a smooth solution, then, from the equation we can get an optimal investment strategy, at last, we gain some conclusions when the claim size is heavy-tailed: if 1-F is regularly varying with indexρ＜-1,then the ruin probabilityψis also regularly varying with indexρ＜-1, and for subexponentially distributed claim sizes, we find the asymptotic behavior of the ruin probabilityψas well as the investment function A.