Rare-Event Simulation for Markov-Modulated Perpetuities
Seminar Room 1, Newton Institute
AbstractIn this talk I will discuss two rare-event simulation problems with both heavy-tailed and Markovmodulated features. Both of which involve the use of Lyapunov inequality and state-dependent sampling that relies crucially on the regenerative feature of the modulating process. The first problem is the computation of large deviations probability of a perpetuity in which both the cash flow and discount rate (can be positive or negative and unbounded) are modulated by underlying Markov economic environment. Despite our assumption that discount rate possesses finite exponential moment, the functional form of perpetuity in terms of discount rate leads to power-law decay. The problem is interesting from a simulation standpoint because of the occurrence of both light and heavy-tailed features: the existence of exponential moment allows us to perform exponential tilting on the discount rate, yet the power-law decay suggests the one-big-jump intuition of heavytailed processes and hence a careful Lyapunov-type analysis is required to guarantee asymptotic optimality. The second problem that we consider is the computation of the first passage probability for a Markov-modulated random walk. In this case a generalized version of Lyapunov inequality is considered, through breaking down the random walk into regenerative cycles. Asymptotic optimality of the algorithm is obtained by careful tuning of parameters that depend on the initial state of the current cycle as well as the current state of the walk.
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