This is an investigation of a new importance sampling technique called twisted simulation. In the i.i.d. case it corresponds to simulating the events of interest via an exponentially shifted version of the direct simulation probability measures. In the finite dimensional Markov chains of i.i.d. random variables the method is optimal over a large class of importance sampling distributions in the sense of minimizing the asymptotic variance of the estimates. The work may be extended to more useful probability models, in particular Markov additive processes and multidimensional event sets. The theoretical extensions require different analysis techniques than have been used previously. The techniques and tools involved will be sophisticated and involved, bust due to the conjectured optimality of the method, the effort expended on these theoretical investigations should have substantial payoff in applications. The methods promise to provide a rigorous structure for the pursuit of the study of simulation of systems with memory.
