Abstract This paper considers ruin problems for insurance companies investing their reserves in stochastic volatility financial markets driven by hidden Markov chains in the framework of Sparre–Andersen insurance models. Since for such models the corresponding renewal equation has dependent coefficients and, as a consequence, the crucial conditions of the Goldie theorem are not satisfied, we develop here a drastically different approach to solve the problem, based on the renewal method proposed by Klüppelberg and Pergamenchtchikov (2004) for general Markov models. On the basis of this result, the asymptotic form for the ruin probability is obtained. More precisely, we find sufficient conditions under which it is shown that the ruin probability decays to zero with the power rate as the initial capital goes to infinity. Next, we construct financial strategies for which this convergence rate can be calculated explicitly; it has been established that this rate can be made more rapid than any power rate. Finally, Monte Carlo simulations which confirm the obtained theoretical results are given.
Pchelintsev et al. (Tue,) studied this question.