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In this article, we study a renewal risk model with compound dependence structures and stochastic returns. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential Lévy process. Assume that claim sizes follow a one-sided linear process with independent and identically distributed steps sizes, and the step sizes and the corresponding inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. By restricting the distribution of the step sizes to the class of extended regular variation (), we obtain some asymptotic estimates, which holds uniformly for all time horizons. Finally, we show the accuracy of the derived asymptotic formula by numerical simulations.

Uniform asymptotics for ruin probabilities of a time-dependent renewal risk model with dependence structures and stochastic returns