Computing Rare Event Probabilities For Affine Models And General State Space Markov Processes
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Computing Rare-event Probabilities for Affine Models and General State Space Markov Processes
Rare-event simulation concerns computing small probabilities, i.e. rare-event probabilities. This dissertation investigates efficient simulation algorithms based on importance sampling for computing rare-event probabilities for different models, and establishes their efficiency via asymptotic analysis. The first part discusses asymptotic behavior of affine models. Stochastic stability of affine jump diffusions are carefully studied. In particular, positive recurrence, ergodicity, and exponential ergodicity are established for such processes under various conditions via a Foster-Lyapunov type approach. The stationary distribution is characterized in terms of its characteristic function. Furthermore, the large deviations behavior of affine point processes are explicitly computed, based on which a logarithmically efficient importance sampling algorithm is proposed for computing rare-event probabilities for affine point processes. The second part is devoted to a much more general setting, i.e. general state space Markov processes. The current state-of-the-art algorithm for computing rare-event probabilities in this context heavily relies on the solution of a certain eigenvalue problem, which is often unavailable in closed form unless certain special structure is present (e.g. affine structure for affine models). To circumvent this difficulty, assuming the existence of a regenerative structure, we propose a bootstrap-based algorithm that conducts the importance sampling on the regenerative cycle-path space instead of the original one-step transition kernel. The efficiency of this algorithm is also discussed.
Computing Rare-event Probabilities for Affine Models and General State Space Markov Processes
Rare-event simulation concerns computing small probabilities, i.e. rare-event probabilities. This dissertation investigates efficient simulation algorithms based on importance sampling for computing rare-event probabilities for different models, and establishes their efficiency via asymptotic analysis. The first part discusses asymptotic behavior of affine models. Stochastic stability of affine jump diffusions are carefully studied. In particular, positive recurrence, ergodicity, and exponential ergodicity are established for such processes under various conditions via a Foster-Lyapunov type approach. The stationary distribution is characterized in terms of its characteristic function. Furthermore, the large deviations behavior of affine point processes are explicitly computed, based on which a logarithmically efficient importance sampling algorithm is proposed for computing rare-event probabilities for affine point processes. The second part is devoted to a much more general setting, i.e. general state space Markov processes. The current state-of-the-art algorithm for computing rare-event probabilities in this context heavily relies on the solution of a certain eigenvalue problem, which is often unavailable in closed form unless certain special structure is present (e.g. affine structure for affine models). To circumvent this difficulty, assuming the existence of a regenerative structure, we propose a bootstrap-based algorithm that conducts the importance sampling on the regenerative cycle-path space instead of the original one-step transition kernel. The efficiency of this algorithm is also discussed.
Random Processes for Engineers
Author: Bruce Hajek
language: en
Publisher: Cambridge University Press
Release Date: 2015-03-12
An engaging introduction to the critical tools needed to design and evaluate engineering systems operating in uncertain environments.