Although stochastic models driven by latent Markov processes are widely used, the classical importance sampling method based on the exponential tilting method for these models suffers from the difficulty of computing the eigenvalue and associated eigenfunction and the plausibility of the indirect asymptotic large deviation regime for the variance of the estimator. We propose a general importance sampling framework that twists the observable and latent processes separately based on a link function that directly minimizes the estimator's variance. An optimal choice of the link function is chosen within the locally asymptotically normal family. We show the logarithmic efficiency of the proposed estimator under the asymptotic normal regime. As applications, we estimate an overflow probability under a pandemic model and the CoVaR, a measurement of the co-dependent financial systemic risk. Both applications are beyond the scope of traditional importance sampling methods due to their nonlinear structures.

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