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Sampling per Mode for Rare Event Simulation in Switching Diffusions
Abstract
A straightforward application of an interacting particle system to estimate a rare event for switching diffusions fails to produce reasonable estimates within a reasonable amount of simulation time. To overcome this, a conditional “sampling per mode†algorithm has been proposed by Krystul in [10]; instead of starting the algorithm with particles randomly distributed, we draw in each mode, a fixed number particles and at each resampling step, the same number of particles is sampled for each visited mode. In this paper, we establish a law of large numbers as well as a central limit theorem for the estimate.
... This Feynman-Kac setting subsequently supported the evaluation of the reach probability through sequential Monte Carlo simulation in the form of an Interacting Particle System (IPS), including proof of convergence [11]. Krystul et al. [28] have used the Feynman-Kac setting to prove convergence of IPS using sampling per mode for a switching diffusion. ...
For diffusions, a well-developed approach in rare event estimation is to introduce a suitable factorization of the reach probability and then to estimate these factors through simulation of an Interacting Particle System (IPS). This paper studies IPS based reach probability estimation for General Stochastic Hybrid Systems (GSHS). The continuous-time executions of a GSHS evolve in a hybrid state space under influence of combinations of diffusions, spontaneous jumps and forced jumps. In applying IPS to a GSHS, simulation of the GSHS execution plays a central role. From literature, two basic approaches in simulating GSHS execution are known. One approach is direct simulation of a GSHS execution. An alternative is to first transform the spontaneous jumps of a GSHS to forced transitions, and then to simulate executions of this transformed version. This paper will show that the latter transformation yields an extra Markov state component that should be treated as being unobservable for the IPS process. To formally make this state component unobservable for IPS, this paper also develops an enriched GSHS transformation prior to transforming spontaneous jumps to forced jumps. The expected improvements in IPS reach probability estimation are also illustrated through simulation results for a simple GSHS example.
... More recently, an IPS (Interacting Particle System) method (a.k.a. Genetic Genealogical algorithm) has been designed in [Del04,DG05,CDLL06] to estimate the probability of rare event related to the terminal value of a Markov chain, see [KLL12] for switching di↵usions. And in [CDFG12], the IPS algorithm is used to estimate probability of rare event related to a static finite dimensional distribution via a particular Markov kernel, which is called shaking transformation in this work. ...
We introduce random transformations called reversible shaking transformations which we use to design two schemes for estimating rare event probability. One is based on interacting particle systems (IPS) and the other on time-average on a single path (POP) using ergodic theorem. We discuss their convergence rates and provide numerical experiments including continuous stochastic processes and jump processes. Our examples cover rather important situations related to insurance, queueing system and random graph for instance. Both schemes have good performance, with a seemingly better one for POP.
This article deals with estimations of probabilities of rare events
using fast simulation based on the splitting method. In this technique,
the sample paths are split into multiple copies at various stages in the
simulation. Our aim is to optimize the algorithm and to obtain a precise
confidence interval of the estimator using branching processes. The
numerical results presented suggest that the method is reasonably
efficient.
We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman–Kac models. We provide an original stochastic analysis-based on Feynman–Kac semigroup techniques combined with recently developed coa-lescent tree-based functional representations of particle block distributions. We present some regularity conditions under which the L 2 -relative error of these weighted particle measures grows linearly with respect to the time horizon yielding what seems to be the first results of this type for this class of unnormalized models. We also illustrate these results in the context of particle absorption models, with a special interest in rare event analysis. Résumé. Nous présentons un théorème non asymptotique pour les approximation par systèmes de particules en interaction des modèles de Feynman–Kac non normalisés. Nous introduisons une analyse stochastique originale basée sur des techniques de semigroupes de Feynman–Kac, associées avec les représentation, récemment proposées, des distributions de blocks de particules, en terme de développement en arbre de coalescence. Nous présentons des conditions de régularité sous lesquelles l'erreur relative L 2 de ces mesures particulaires pondérées croît linéairement par rapport 'a l'horizon temporel, conduisant 'a ce qui semble être le premier résultat de ce type pour cette classe de modèles non normalisés. Nous illustrons ces résultats dans le contexte des mesures statiques de Boltzmann–Gibbs et des distributions restreintes, avec un intéret partuculier pour les événements rares.
IntroductionStatic problemsMarkov chainsAlgorithmsReferences
IntroductionPrinciples and implementationsAnalysis in a simplified setting: a coin-flipping modelAnalysis and central limit theorem in a more general settingA numerical illustrationReferences
In response to the increasing needs for control and optimization of hybrid systems, this work is concerned with such asymptotic properties as recurrence (also known as weak stochastic stability in the literature) and ergodicity of regime-switching diffusions. Using Liapunov functions, necessary and sufficient conditions for positive recurrence are developed. Then, ergodicity of positive recurrent regime-switching diffusions is obtained by constructing cycles using the associated discrete-time Markov chains.
We present in this article a genetic type interacting particle systems algorithm and a genealogical model for estimating a class of rare events arising in physics and network analysis. We represent the distribution of a Markov process hitting a rare target in terms of a Feynman–Kac model in path space. We show how these branching particle models described in previous works can be used to estimate the probability of the corresponding rare events as well as the distribution of the process in this regime.
This paper presents a method for fast simulation of rare events, called RESTART (REpetitive Simulation Trials After Reaching Thresholds). The method obtains dramatic computer time savings for an equal confidence of the results. The paper describes the method in its multiple thresholds version, shows the efficiency of the method and gives criteria to choose its parameters. It presents a simulation library (ASTRO) in which RESTART is implemented and, finally, application examples to teletraffic and reliability problems are provided.
Embedding of rare event estimation theory within a stochastic analysis framework has recently led to significant novel results in rare event estimation for a diffusion process using sequential MC simulation. This chapter presents this rare event estimation theory for diffusions to a Stochastic Hybrid System (SHS) and extends it in order to handle a large scale SHS where a very huge number of rare discrete modes may contribute significantly to the rare event estimation. Essentially, the approach taken is to introduce a suitable aggregation of the discrete modes, and to develop importance sampling and Rao-Blackwellization relative to these aggregated modes. The practical use of this approach is demonstrated for the estimation ofmid-air collision for an advanced air traffic control example.
Stochastic dynamical modelling of accident risk is of high interest for the safe design of complex safety-critical systems and operations, such as in nuclear and chemical industries, and advanced air traffic management. In comparison with statistical analysis of collected data, stochastic dynamical modelling approach has the advantage of enabling the use of stochastic analysis and advanced Monte Carlo simulation approaches.
Switching Diffusion processes can be represented as pathwise unique solutions of SDE's in a hybrid state space that are driven by Brownian motion and Poisson random measure. This paper extends these SDE's to switching jump- diffusions, the jumps of which i) happen simultaneously with mode switching, and ii) depend on the mode after the switching. Jumps satisfying both i) and ii) are referred to as hybrid jumps. Because of ii) there is an anticipation effect in the SDE, which makes the hybrid jump extension challenging. Copyright ‚, 2003, IFAC.
A sequential particle algorithm proposed by Oudjane (2000) is studied here, which uses an adaptive random number of particles
at each generation and guarantees that the particle system never dies out. This algorithm is especially useful for approximating
a nonlinear (normalized) Feynman-Kac flow, in the special case where the selection functions can take the zero value, e.g.
in the simulation of a rare event using an importance splitting approach. Among other results, a central limit theorem is proved by induction, based on the result of Rényi (1957) for sums
of a random number of independent random variables. An alternate proof is also given, based on an original central limit theorem
for triangular arrays of martingale increments spread across generations with different random sizes.
This article focuses on branching particle interpretations of rare events. We connect importance sampling techniques with
interacting particle algorithms, and multi-splitting branching models. These Monte Carlo methods are illustrated with a variety
of examples arising in particle trapping analysis, as well as in ruin type estimation problems. We also provide a rather detailed
presentation of the asymptotic theory of these particle algorithms, including exponential extinction probabilities,
\mathbbLp\mathbb{L}_p
-mean error bounds, central limit theorem, and fluctuation variance comparaisons.
RESTART (Repetitive Simulation Trials After Reaching Thresholds) is a widely applicable accelerated simulation technique that allows the evaluation of extremely low probabilities. This paper revisits the theoretical basis of RESTART in a more general and rigorous way. The unbiasedness of the estimator is proved and its variance is derived without need of making any assumption on the simulated system. From this analysis, the gain obtained with RESTART as well as the optimal parameter values that maximise the gain are derived. The sensitivity of the method to the choice of the importance function, threshold values and number of retrials for each threshold is studied, providing simple bounds on me impact on the gain of each of these choices.
RESTART (Repetitive Simulation Trials After Reaching Thresholds) is a widely applicable accelerated simulation technique that allows the evaluation of extremely low probabilities. The focus of this article is on providing guidelines for achieving a high efficiency in a simulation with RESTART. Emphasis is placed on the choice of the importance function, that is, the function of the system state for determining when retrials are made. A heuristic approach which is shown to be effective for some systems is proposed for this choice. A two-queue tandem network is used to illustrate the efficiency achieved following these guidelines. The importance function chosen in this example shows that an appropriate choice of the importance function leads to an efficient simulation of a system with multidimensional state space. Also presented are sufficient conditions for achieving asymptotic efficiency, and it is shown that they are not very restrictive in RESTART simulation.
The past fifty years the field of the estimation of rare event probabilities has grown considerably, partly because of the enormous growth in computing power during this period. Because it still is and will not ever be possible to estimate these probabilities efficiently using standard techniques a multitude of methods has been developed and described in the literature and incorporated into simulation packages. The single two most important techniques are based on simple principles. Importance Sampling Change the system in a way that makes the probabilities to estimate become large, so that standard methods can be applied again. One can get the original probabilities back by accounting for the system transformation used. Importance Splitting Change the paths traversed in the simulation in a way that the promising paths are split into a multitude of lightweight paths. In this manner one obtains more activity in the interesting area and one will see the rare event happening more frequently, making the estimates better. Both techniques are about fifty years old and have evolved in several directions. Here we have limited the research to the latter method. The splitting method has been reinvented a number of times in the literature and it has only reached maturity during the last decade. Before, the method was limited to simple models and it was not asymptotically efficient. In this thesis a mathematical foundation is developed for the used methods; the efficiency and complexity of the basic algorithm are also derived. New techniques are developed and compared based on efficiency measures on a broad range of reference models. We see that all known problems and limitations can be dealt with using new techniques that enrich the splitting method. The combination of the analytical approach of the proposed methods and the validation in practice produces a strategy whose efficiency is optimal for a broad class of models and problems. The collection of methods and techniques is gathered in a tool designed explicitly to solve a broad range of rare event problems. The practical application of the work reported here in this thesis can be found in modern communication networks where one is interested in the quality of service delivered to the customer. The (hopefully) rare event in such a setting is the probability of loss of data which the customer wishes to transfer over the network. The presented method will apply to many rare event problems; the currently most common practical use for the method is probably the telecommunications area.
We consider the stochastic processes Xk+1 = [Gamma]k+1(Xk) + Wk+1 where {[Gamma]k} is a sequence of nonlinear random functions and {Wk} is a sequence of disturbances. Sufficient conditions for the existence of a unique invariant probability are obtained. Functional central limit theorem is proved for every Lipschitzian function on R.
ying at least the rough asymptotics of a rare event probability, often described by a large deviations result. This type of analysis can be formidable in complex models, so the domain of importance sampling, while substantial, does not include all problems of interest. This work deals with an alternative method for rare event simulation that uses the technique of splitting # Also a#liated to IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 sample paths. The main advantage of this technique is that it appears to require rather little model structure for its applicability. Splitting for rare event simulation was originally discussed by #6# in the context of estimating rare particle transmission probabilities in physics. Since then, there were only a few intermittent references to the use of this technique for rare event simulation ##2#, #1#, #5##. However, recently it was revisited in a signi#cantwayby#9#, #8#, and #10# for estimating probabilities of r
Rare event estimation for a large-scale stochastic hybrid system with air traffic application
- Henk Blom
- Bert Bakker
- Jaroslav Krystul
Henk Blom, Bert Bakker, and Jaroslav Krystul. Rare event estimation for a large-scale stochastic hybrid
system with air traffic application. In Gerardo Rubino and Bruno Tuffin, editors, Rare Event Simulation
using Monte Carlo Methods, pages 194-214. Wiley, 2009.
Splitting techniques
- François Le Pierre L'écuyer
- Pascal Gland
- Bruno Lezaud
- Tuffin
Pierre L'Écuyer, François Le Gland, Pascal Lezaud, and Bruno Tuffin. Splitting techniques. In Gerardo
Rubino and Bruno Tuffin, editors, Rare Event Simulation using Monte Carlo Methods, pages 39-61. Wiley,
2009.