Cox risk model with variable premium rate and stochastic return on investment

This paper studies the ruin probability for a Cox risk model with intensity depending on premiums and stochastic investment returns, and the model proposed in this paper allows the dependence between premiums and claims. When the surplus is invested in the bond market with constant interest force, coupled integral equations for the Gerber–Shiu expected discounted penalty function (GS function) are derived; together with the initial value and Laplace transformation of the GS function, we provide a numerical procedure for obtaining the GS function. When the surplus can be invested in risky asset driven by a drifted Brownian motion, we focus on finding a minimal upper bound of ruin probability and find that optimal piecewise constant policy yields the minimal upper bound. It turns out that the optimal piecewise constant policy is asymptotically optimal when initial surplus tends to infinity.