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.