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Rare event simulation for processes generated via stochastic fixed point equations. (English) Zbl 1316.65015
A simulation algorithm of estimating the tail probability of a random variable \(V\), satisfying an equation \(V = f(V)\) in the sense of distribution, is considered, where \(f(v)=A\max(v,D)+B\), \(A,B,D\) are random variables. The stochastic fixed point problem in the sense of law typically comes from, for instance, the financial time series, the ruin problem with stochastic investments, ARCH(1) and so on. A dynamic importance sampling estimator is given, which is shown to be consistent, strongly efficient and optimal. The running time of the algorithm is described. Numerical examples are discussed.

MSC:
65C50 Other computational problems in probability (MSC2010)
65C05 Monte Carlo methods
91G60 Numerical methods (including Monte Carlo methods)
60H25 Random operators and equations (aspects of stochastic analysis)
60F10 Large deviations
60G40 Stopping times; optimal stopping problems; gambling theory
60J05 Discrete-time Markov processes on general state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 Computational methods in Markov chains
60K15 Markov renewal processes, semi-Markov processes
65C40 Numerical analysis or methods applied to Markov chains
91B30 Risk theory, insurance (MSC2010)
91B84 Economic time series analysis
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