On asymptotically efficient simulation of ruin probabilities in a Markovian environment

Lehtonen, Tapani; Nyrhinen, Harri

doi:10.1080/03461238.1992.10413897

Cited by 32 publications

(14 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Þrst result of this type was given by Siegmund [37]. The basic idea was subsequently investigated in many contexts, and a small selection of the literally hundreds of papers on the topics is [1,2,3,7,8,9,11,12,14,23,25,28,29,31,32,35]. Necessary and sufficient conditions under which a prescribed scheme is asymptotically optimal are discussed in [10,33,34], while [26] gives a survey of rare-event simulation.…”

Section: Introductionmentioning

confidence: 99%

Importance Sampling, Large Deviations, and Differential Games

Dupuis¹,

Wang²

2002

161

View full text Add to dashboard Cite

A heuristic that has emerged in the area of importance sampling is that the changes of measure used to prove large deviation lower bounds give good performance when used for importance sampling. Recent work, however, has suggested that the heuristic is incorrect in many situations. The perspective put forth in the present paper is that large deviation theory suggests many changes of measure, and that not all are suitable for importance sampling. In the setting of Cramér's Theorem, the traditional interpretation of the heuristic suggests a Þxed change of distribution on the underlying independent and identically distributed summands. In contrast, we consider importance sampling schemes where the exponential change of measure is adaptive, in the sense that it depends on the historical empirical mean. The existence of asymptotically optimal schemes within this class is demonstrated. The result indicates that an adaptive change of measure, rather than a static change of measure, is what the large deviations analysis truly suggests. The proofs utilize a control-theoretic approach to large deviations, which naturally leads to the construction of asymptotically optimal adaptive schemes in terms of a limit Bellman equation. Numerical examples contrasting the adaptive and standard schemes are presented, as well as an interpretation of their different performances in terms of differential games. * Research of this author supported in part by the National Science Foundation (NSF-DMS-0072004, NSF-ECS-9979250) and the Army Research Office (DAAD19-00-1-0549, DAAD19-02-1-0425).† Research of this author supported in part by the National Science Foundation (NSF-DMS-0103669). Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

show abstract

Section: Introductionmentioning

confidence: 99%

Importance Sampling, Large Deviations, and Differential Games

Dupuis¹,

Wang²

2002

161

View full text Add to dashboard Cite

show abstract

“….. For the estimation of this probability, Lehtonen and Nyrhinen (1992b) show the asymptotic efficiency of the exponential twist in the regime M → ∞; they also establish uniqueness properties of the exponential twist. Related results in a more general Markovian setting are obtained by Asmussen (1989) and Lehtonen and Nyrhinen (1992a). Collamore (2002) extends this to a multi-dimensional setting.…”

Section: Introductionmentioning

confidence: 82%

On asymptotically efficient simulation of large deviation probabilities

Dieker

Mandjes

2005

Advances in Applied Probability

View full text Add to dashboard Cite

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a PNA Probability, Networks and Algorithms Probability, Networks and Algorithms ABSTRACTConsider a family of probabilities for which the decay is governed by a large deviation principle. To find an estimate for a fixed member of this family, one is often forced to use simulation techniques. Direct Monte Carlo simulation, however, is often impractical, particularly if the probability that should be estimated is extremely small. Importance sampling is a technique in which samples are drawn from an alternative distribution, and an unbiased estimate is found after a likelihood ratio correction. Specific exponentially twisted distributions were shown to be good sampling distributions under fairly general circumstances. In this paper, we present necessary and sufficient conditions for asymptotic efficiency of a single exponentially twisted distribution, sharpening previously established conditions. Using the insights that these conditions provide, we construct an example for which we explicitly compute the 'best' change of measure. However, simulation using the new measure faces exactly the same difficulties as direct Monte Carlo simulation. We discuss the relation between this example and other counterexamples in the liturature.We also apply the conditions to find necessary and sufficient conditions for asymptotic efficiency of the exponential twist in a Mogul skiȋ sample-path problem. An important special case of this problem is the probability of ruin within finite time.

show abstract

“…Large deviation results for MAPs can be found in papers of Iscoe et al [22], Ney and Nummelin [31,32] and Lehtonen and Nyrhinen [25]. Early applications of these processes used them successfully as models of packetized voice data, see, for example, Heffes and Lucantoni [21].…”

Section: Modulated Processesmentioning

confidence: 91%

On the large deviations of a class of modulated additive processes

2011

View full text Add to dashboard Cite

Abstract. We prove that the large deviation principle holds for a class of processes inspired by semiMarkov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated Lévy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union's standardized stochastic model of two-way conversational speech.Mathematics Subject Classification. 60F10, 60K15.

show abstract

On asymptotically efficient simulation of ruin probabilities in a Markovian environment

Cited by 32 publications

References 15 publications

Importance Sampling, Large Deviations, and Differential Games

Importance Sampling, Large Deviations, and Differential Games

On asymptotically efficient simulation of large deviation probabilities

On the large deviations of a class of modulated additive processes

Contact Info

Product

Resources

About