7 - Risk Model with Variable Premium Intensity and Investments in One Risky Asset

Abstract

In this chapter, we consider a generalization of the classical risk model where 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. Our main aim is to show that if the premium intensity grows rapidly with increasing surplus, then an upper exponential bound for the ruin probability holds under certain conditions in spite of the fact that all surplus is invested in the risky asset. To this end, we apply the supermartingale approach and allow the surplus process to explode. To be more precise, we let the premium intensity be a quadratic function. In addition, we investigate the question concerning the probability of explosion of the surplus process between claim arrivals in detail.