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Subsolutions of an Isaacs equation and efficient schemes for importance sampling: Convergence analysis
, 2005
"... It was established in [6, 7] that importance sampling algorithms for estimating rareevent probabilities are intimately connected with twoperson zerosum differential games and the associated Isaacs equation. This game interpretation shows that dynamic or statedependent schemes are needed in orde ..."
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Cited by 51 (18 self)
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It was established in [6, 7] that importance sampling algorithms for estimating rareevent probabilities are intimately connected with twoperson zerosum differential games and the associated Isaacs equation. This game interpretation shows that dynamic or statedependent schemes are needed in order to attain asymptotic optimality in a general setting. The purpose of the present paper is to show that classical subsolutions of the Isaacs equation can be used as a basic and flexible tool for the construction and analysis of efficient dynamic importance sampling schemes. There are two main contributions. The first is a basic theoretical result characterizing the asymptotic performance of importance sampling estimators based on subsolutions. The second is an explicit method for constructing classical subsolutions as a mollification of piecewise affine functions. Numerical examples are included for illustration and to demonstrate that simple, nearly asymptotically optimal importance sampling schemes can be obtained for a variety of problems via the subsolution approach.
Dynamic importance sampling for queueing networks
, 2005
"... Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a prio ..."
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Cited by 39 (11 self)
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Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a twonode tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include dnode tandem Jackson networks and a twonode network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers.
P.W.: Efficient rareevent simulation for the maximum of heavytailed random walks
 Annals of Applied Probability
, 2008
"... Let (Xn:n ≥ 0) be a sequence of i.i.d. r.v.’s with negative mean. Set S0 = 0 and define Sn = X1 + · · ·+Xn. We propose an importance sampling algorithm to estimate the tail of M = max{Sn:n ≥ 0} that is strongly efficient for both light and heavytailed increment distributions. Moreover, in the case ..."
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Cited by 31 (15 self)
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Let (Xn:n ≥ 0) be a sequence of i.i.d. r.v.’s with negative mean. Set S0 = 0 and define Sn = X1 + · · ·+Xn. We propose an importance sampling algorithm to estimate the tail of M = max{Sn:n ≥ 0} that is strongly efficient for both light and heavytailed increment distributions. Moreover, in the case of heavytailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is statedependent. In the presence of light tails, our procedure leads to Siegmund’s (1979) algorithm. The rigorous analysis of efficiency requires new Lyapunovtype inequalities that can be useful in the study of more general importance sampling algorithms. 1. Introduction. In
Dynamic importance sampling for uniformly recurrent markov chains
 Annals of Applied Probability
, 2005
"... Importance sampling is a variance reduction technique for efficient estimation of rareevent probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, howe ..."
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Cited by 27 (6 self)
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Importance sampling is a variance reduction technique for efficient estimation of rareevent probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper we consider dynamic importance sampling in the setting of uniformly recurrent Markov chains. By “dynamic ” we mean that in the course of a single simulation, the change of measure can depend on the outcome of the simulation up till that time. Based on a controltheoretic approach to large deviations, the existence of asymptotically optimal dynamic schemes is demonstrated in great generality. The implementation of the dynamic schemes is carried out with the help of a limiting Bellman equation. Numerical examples are presented to contrast the dynamic and standard schemes. 1. Introduction. Among
Importance sampling for sums of random variables with regularly varying tails. Working paper
, 2006
"... Importance sampling is a variance reduction technique for efficient estimation of rareevent probabilities by Monte Carlo. For random variables with heavy tails there is little consensus on how to choose the change of measure used in importance sampling. In this paper we study dynamic importance sam ..."
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Cited by 26 (4 self)
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Importance sampling is a variance reduction technique for efficient estimation of rareevent probabilities by Monte Carlo. For random variables with heavy tails there is little consensus on how to choose the change of measure used in importance sampling. In this paper we study dynamic importance sampling schemes for sums of independent and identically distributed random variables with regularly varying tails. The number of summands can be random but must be independent of the summands. For estimating the probability that the sum exceeds a given threshold, we explicitly identify a class of dynamic importance sampling algorithms with bounded relative errors. In fact, these schemes are nearly asymptotically optimal in the sense that the second moment of the corresponding importance sampling estimator can be made as close as desired to the minimal possible value. 1
FAST SIMULATION FOR MULTIFACTOR PORTFOLIO CREDIT RISK IN THE tCOPULA MODEL
, 2005
"... We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the tcopula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and ..."
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Cited by 22 (2 self)
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We present an importance sampling procedure for the estimation of multifactor portfolio credit risk for the tcopula model, i.e, the case where the risk factors have the multivariate t distribution. We use a version of the multivariate t that can be expressed as a ratio of a multivariate normal and a scaled chisquare random variable. The procedure consists of two steps. First, using the large deviations result for the Gaussian model in Glasserman, Kang, and Shahabuddin (2005a), we devise and apply a change of measure to the chisquare random variable. Then, conditional on the chisquare random variable, we apply the importance sampling procedure developed for the Gaussian copula model in Glasserman, Kang, Shahabuddin (2005b). We support our importance sampling procedure by numerical examples.
Importance sampling for Jackson networks
, 2008
"... Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asympt ..."
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Cited by 15 (3 self)
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Rare event simulation in the context of queueing networks has been an active area of research for more than two decades. A commonly used technique to increase the efficiency of Monte Carlo simulation is importance sampling. However, there are few rigorous results on the design of efficient or asymptotically optimal importance sampling schemes for queueing networks. Using a recently developed game/subsolution approach, we construct simple and efficient statedependent importance sampling schemes for simulating buffer overflows in stable open Jackson networks. The sampling distributions do not depend on the particular event of interest, and hence overflow probabilities for different events can be estimated simultaneously. A byproduct of the analysis is the identification of the minimizing trajectory for the calculus of variation problem that is associated with the samplepath large deviation rate function. 1
Rareevent simulation techniques: An introduction and recent advances
 Handbook of Simulation, volume 13 of Handbooks in Operations Research and Management Science
, 2006
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