8 - Risk Model with Variable Premium Intensity and Investments in One Risky Asset up to the Stopping Time of Investment Activity

Abstract

In this chapter, we consider a generalization of the classical risk model where a premium intensity depends on a current surplus of an insurance company. All surplus is invested in one risky asset, the price of which follows a geometric Brownian motion, but an insurance company stops its investment activity when the price of the risky asset goes down below some fixed level. Our main result asserts that an upper exponential bound for the ruin probability holds under certain conditions. To this end, we give another representation for the surplus process, redefine the ruin time and establish the supermartingale property for an auxiliary exponential process. Then, we concentrate on the case of exponentially distributed claim sizes. Moreover, we consider the modification of the model where the insurance company stops its investment activity when the price of the risky asset exits from some fixed interval and extend the results obtained to this modification.