scholarly journals Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012 ◽  
Vol 44 (04) ◽  
pp. 1173-1196
Author(s):  
Hock Peng Chan ◽  
Shaojie Deng ◽  
Tze-Leung Lai
Keyword(s):  
Random Walks ◽  
Importance Sampling ◽  
Rare Event ◽  
Sequential Importance Sampling ◽  
New Approach ◽  
Martingale Representation ◽  
Event Simulation ◽  
Markov Random ◽  
Heavy Tailed ◽  
Asymptotically Efficient

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

scholarly journals Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

2012 ◽  
Vol 44 (4) ◽  
pp. 1173-1196 ◽  
Author(s):  
Hock Peng Chan ◽  
Shaojie Deng ◽  
Tze-Leung Lai
Keyword(s):  
Random Walks ◽  
Importance Sampling ◽  
Rare Event ◽  
Sequential Importance Sampling ◽  
New Approach ◽  
Martingale Representation ◽  
Event Simulation ◽  
Markov Random ◽  
Heavy Tailed ◽  
Asymptotically Efficient

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

Estimation of cyber network risk using rare event simulation

2020 ◽  
pp. 154851292093455
Author(s):  
Alexander L Krall ◽  
Michael E Kuhl ◽  
Shanchieh J Yang
Keyword(s):  
Importance Sampling ◽  
Computational Simulation ◽  
Rare Event ◽  
Rare Event Simulation ◽  
Modeling And Analysis ◽  
Event Simulation ◽  
Importance Sampling Method ◽  
Statistical Inferences ◽  
Analysis Technique ◽  
Network Risk

Inherent vulnerabilities in a cyber network’s constituent machine services can be exploited by malicious agents. As a result, the machines on any network are at risk. Security specialists seek to mitigate the risk of intrusion events through network reconfiguration and defense. When dealing with rare cyber events, high-quality risk estimates using standard simulation approaches may be unattainable, or have significant attached uncertainty, even with a large computational simulation budget. To address this issue, an efficient rare event simulation modeling and analysis technique, namely, importance sampling for cyber networks, is developed. The importance sampling method parametrically amplifies certain aspects of the network in order to cause a rare event to happen more frequently. Output collected under these amplified conditions is then scaled back into the context of the original network to provide meaningful statistical inferences. The importance sampling methodology is tailored to cyber network attacks and takes the attacker’s successes and failures as well as the attacker’s targeting choices into account. The methodology is shown to produce estimates of higher quality than standard simulation with greater computational efficiency.

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Improved algorithms for rare event simulation with heavy tails

2006 ◽  
Vol 38 (2) ◽  
pp. 545-558 ◽  
Author(s):  
Søren Asmussen ◽  
Dirk P. Kroese
Keyword(s):  
Monte Carlo ◽  
Relative Error ◽  
Heavy Tails ◽  
Numerical Study ◽  
Rare Event ◽  
Random Variable ◽  
Event Simulation ◽  
Conditional Monte Carlo ◽  
Heavy Tailed ◽  
Geometric Random Variable

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

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