Optimal rare event Monte Carlo for Markov modulated regularly varying random walks

Most of the efficient rare event simulation methodology for heavy-tailed systems has concentrated on processes with stationary and independent increments. Motivated by applications such as insurance risk theory, in this paper we develop importance sampling estimators that are shown to achieve asymptotically vanishing relative error property (and hence are strongly efficient) for the estimation of large deviation probabilities in Markov modulated random walks that possess heavy-tailed increments. Exponential twisting based methods, which are effective in light-tailed settings, are inapplicable even in the simpler case of random walk involving i.i.d. heavy-tailed increments. In this paper we decompose the rare event of interest into a dominant and residual component, and simulate them independently using state-independent changes of measure that are both intuitive and easy to implement.