Stochastic Simulation: Algorithms and Analysis

Asmussen, Søren; Glynn, Peter W.

doi:10.1007/978-0-387-69033-9

Cited by 1,013 publications

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“…Moreover, an estimator is strongly efficient or is said to have bounded relative error if sup b≥0 EZ 2 b /α(b) 2 < ∞. These notions are standard; see for instance Asmussen and Glynn (2007), Bucklew (2004), Juneja and Shahabuddin (2006). Finally, a notion that has been recently introduced (see Juneja 2007) is that of asymptotically vanishing relative error, which goes beyond strong efficiency and requires the second moment of the estimator to achieve the best possible asymptotic performance, namely lim b−→∞ EZ 2 b /α(b) 2 = 1.…”

Section: Introductionmentioning

confidence: 98%

Efficient simulation of tail probabilities of sums of correlated lognormals

et al. 2009

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We consider the problem of efficient estimation of tail probabilities of sums of correlated lognormals via simulation. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose two estimators that can be rigorously shown to be efficient as the tail probability of interest decreases to zero. The first estimator, based on importance sampling, involves a scaling of the whole covariance matrix and can be shown to be asymptotically optimal. A further study, based on the Cross-Entropy algorithm, is also performed in order to adaptively optimize the scaling parameter of the covariance. The second estimator decomposes the probability of interest in two contributions and takes advantage of the fact that large deviations for a sum of correlated lognormals are (asymptotically) caused by the largest increment. Importance sampling is then applied to each of these contributions to obtain a combined estimator with asymptotically vanishing relative error.

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Section: Introductionmentioning

confidence: 98%

Efficient simulation of tail probabilities of sums of correlated lognormals

et al. 2009

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“…, X n are independent and identically distributed (iid) random variables. This particular problem has attracted considerable attention due to its relevance in queueing theory and various applications in finance and risk management (see, e.g., Cruz 2002;Asmussen and Glynn 2007). The literature on the case where the random variables are thin-tailed is extensive, but not until recently has substantial progress been made on the heavy-tailed case.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, Rubinstein (2007) comments that importance sampling (IS) with high-dimensional likelihood ratio should not be used in those problems. In view of this, instead of considering IS we develop efficient algorithms for estimating rare-event probabilities in high-dimensional settings based on another popular variance reduction idea, namely, conditional Monte Carlo (Asmussen and Glynn 2007;Rubinstein and Kroese 2007). By utilizing an asymptotic description of how the rare event occurs, we derive algorithms that involve generating random variables only from the nominal distributions, thus avoiding the problem of choosing a suitable proposal density.…”

Section: Introductionmentioning

confidence: 99%

“…Although the screening method (Rubinstein 2007) is especially developed for such a setting, it can only mitigate the degeneracy problem. The second general setting involves heavy-tailed random variables, in which case the popular exponential change of measure technique (Asmussen and Glynn 2007) cannot be directly applied. In both settings, however, the asymptotic description of the rare event occurrence can be easily exploited to develop efficient conditional Monte Carlo algorithms.…”

Section: Introductionmentioning

confidence: 99%

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Rare-event probability estimation with conditional Monte Carlo

Chan

Kroese

2009

Ann Oper Res

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Estimation of rare-event probabilities in high-dimensional settings via importance sampling is a difficult problem due to the degeneracy of the likelihood ratio. In fact, it is generally recommended that Monte Carlo estimators involving likelihood ratios should not be used in such settings. In view of this, we develop efficient algorithms based on conditional Monte Carlo to estimate rare-event probabilities in situations where the degeneracy problem is expected to be severe. By utilizing an asymptotic description of how the rare event occurs, we derive algorithms that involve generating random variables only from the nominal distributions, thus avoiding any likelihood ratio. We consider two settings that occur frequently in applied probability: systems involving bottleneck elements and models involving heavytailed random variables. We first consider the problem of estimating P(X 1 + · · · + X n > γ ), where X 1 , . . . , X n are independent but not identically distributed (ind) heavy-tailed random variables. Guided by insights obtained from this model, we then study a variety of more general settings. Specifically, we consider a complex bridge network and a generalization of the widely popular normal copula model used in managing portfolio credit risk, both of which involve hundreds of random variables. We show that the same conditioning idea, guided by an asymptotic description of the way in which the rare event happens, can be used to derive estimators that outperform existing ones.

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“…To simplify the exposition we assume that the state space of the process {X} is discrete. It is possible to drop this restriction, see (Asmussen and Glynn 2007) and the references therein for more details. Suppose that a stationary measure exists as well as a positive recurrent state O.…”

Section: Estimating Stationary Measuresmentioning

confidence: 99%

The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation

Dean

Dupuis

2009

Ann Oper Res

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We consider a general class of branching methods with killing for the estimation of rare events. The class includes a number of existing schemes, including RESTART and DPR (Direct Probability Redistribution). A method for the design and analysis is developed when the quantity of interest can be embedded in a sequence whose limit is determined by a large deviation principle. A notion of subsolution for the related calculus of variations problem is introduced, and two main results are proved. One is that the number of particles and the total work scales subexponentially in the large deviation parameter when the branching process is constructed according to a subsolution. The second is that the asymptotic performance of the schemes as measured by the variance of the estimate can be characterized in terms of the subsolution. Some examples are given to demonstrate the performance of the method.

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Stochastic Simulation: Algorithms and Analysis

Cited by 1,013 publications

References 0 publications

Efficient simulation of tail probabilities of sums of correlated lognormals

Efficient simulation of tail probabilities of sums of correlated lognormals

Rare-event probability estimation with conditional Monte Carlo

The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation

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