Optimal investment of a time-dependent renewal risk model with stochastic return

Consider an insurance company which is allowed to invest into a riskless and a risky asset under a constant mix strategy. The total claim amount is modeled by a non-standard renewal risk model with dependence between the claim size and the inter-arrival time introduced by a Farlie-Gumbel-Morgenstern copula. The price of the risky asset is described by an exponential Lévy process. Based on some known results, the uniform asymptotic estimate for ruin probability with investment strategy is obtained with regularly varying tailed claims. Applying the asymptotic formula, we provide an approximation of the optimal investment strategy to maximize the expected terminal wealth subject to a risk constraint on the Value-at-Risk, which is defined with respect to finite-time discounted net loss. A numerical example is illustrated for the results, which demonstrates that big dependence parameter is advantageous for the insurer. We explain the reason by some inequalities.