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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.

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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[1] Alsmeyer, G., Iksanov, A. and Rösler, U. (2009). On distributional properties of perpetuities. J. Theoret. Probab. 22 666-682. · Zbl 1173.60309 · doi:10.1007/s10959-008-0156-8
[2] Asmussen, S. (2003). Applied Probability and Queues , 2nd ed. Springer, New York. · Zbl 1029.60001
[3] Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation : Algorithms and Analysis . Springer, New York. · Zbl 1126.65001
[4] Blanchet, J., Lam, H. and Zwart, B. (2012). Efficient rare-event simulation for perpetuities. Stochastic Process. Appl. 122 3361-3392. · Zbl 1254.65019 · doi:10.1016/j.spa.2012.05.002
[5] Collamore, J. F. (2002). Importance sampling techniques for the multidimensional ruin problem for general Markov additive sequences of random vectors. Ann. Appl. Probab. 12 382-421. · Zbl 1021.65003 · doi:10.1214/aoap/1015961169
[6] Collamore, J. F. (2009). Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann. Appl. Probab. 19 1404-1458. · Zbl 1176.60018 · doi:10.1214/08-AAP584
[7] Collamore, J. F. and Vidyashankar, A. N. (2013a). Large deviation tail estimates and related limit laws for stochastic fixed point equations. In Random Matrices and Iterated Random Functions (G. Alsmeyer and M. Löwe, eds.) 91-117. Springer, Heidelberg. · Zbl 1280.60040 · doi:10.1007/978-3-642-38806-4_5
[8] Collamore, J. F. and Vidyashankar, A. N. (2013b). Tail estimates for stochastic fixed point equations via nonlinear renewal theory. Stochastic Process. Appl. 123 3378-3429. · Zbl 1292.60070 · doi:10.1016/j.spa.2013.04.015
[9] Collamore, J. F., Vidyashankar, A. N. and Xu, J. (2013). Rare event simulation for stochastic fixed point equations related to the smoothing transform. In Proceedings of the Winter Simulation Conference 555-563.
[10] Dupuis, P. and Wang, H. (2005). Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15 1-38. · Zbl 1068.60036 · doi:10.1214/105051604000001016
[11] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. · Zbl 0724.60076 · doi:10.1214/aoap/1177005985
[12] Heyde, C. C. (1966). Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37 699-710. · Zbl 0143.19102 · doi:10.1214/aoms/1177699465
[13] Iscoe, I., Ney, P. and Nummelin, E. (1985). Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 373-412. · Zbl 0602.60034 · doi:10.1016/0196-8858(85)90017-X
[14] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. · Zbl 0291.60029 · doi:10.1007/BF02392040
[15] Lai, T. L. and Siegmund, D. (1979). A nonlinear renewal theory with applications to sequential analysis. II. Ann. Statist. 7 60-76. · Zbl 0409.62074 · doi:10.1214/aos/1176344555
[16] Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications ( Brunswick , Maine , 1984). Contemp. Math. 50 263-273. Amer. Math. Soc., Providence, RI. · Zbl 0587.60057 · doi:10.1090/conm/050/841098
[17] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators . Cambridge Univ. Press, Cambridge. · Zbl 0551.60066
[18] Siegmund, D. (1976). Importance sampling in the Monte Carlo study of sequential tests. Ann. Statist. 4 673-684. · Zbl 0353.62044 · doi:10.1214/aos/1176343541
[19] Siegmund, D. (1985). Sequential Analysis : Tests and Confidence Intervals . Springer, New York. · Zbl 0573.62071
[20] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783. · Zbl 0417.60073 · doi:10.2307/1426858
[21] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis . SIAM, Philadelphia, PA. · Zbl 0487.62062
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