Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T08:01:02.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonasymptotic performance analysis of importance sampling schemes for small noise diffusions

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 March 2016

Konstantinos Spiliopoulos
Konstantinos Spiliopoulos*
Affiliation:
Boston University
*
∗ Postal address: Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.

Keywords

Importance samplingMonte Carlolarge deviationasymptotic expansion

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 797 - 810
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Blanchet, J. H. (2009). Efficient importance sampling for binary contingency tables. Ann. Appl. Prob. 19 949-982.CrossRefGoogle Scholar
Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18 1351-1378.CrossRefGoogle Scholar
Blanchet, J., Glynn, P. and Leder, K. (2012). On Lyapunov inequalities and subsolutions for efficient importance sampling. ACM Trans. Model. Comput. Simul. 22 27 pp.Google Scholar
Blanchet, J., Glynn, P. and Liu, J. C. (2007). Fluid heuristics, Lyapunov bounds and efficient importance sampling for a heavy-tailed G/G/1 queue. Queueing Systems 57 99-113.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stoch. Stoch. Reports 76 481-508.Google Scholar
Dupuis, P. and Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Operat. Res. 32 723-757.CrossRefGoogle Scholar
Dupuis, P., Spiliopoulos, K. and Wang, H. (2012). Importance sampling for multiscale diffusions. Multiscale Model. Simul. 10 1-27.Google Scholar
Dupuis, P., Spiliopoulos, K. and Zhou, X. (2015). Escaping from an attractor: importance sampling and rest points I. Ann. Appl. Prob. 25 2909-2958.CrossRefGoogle Scholar
Fleming, W. H. and James, M. R. (1992). Asymptotic series and exit time probabilities. Ann. Prob. 20, 1369-384.Google Scholar
Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York.Google Scholar
Fleming, W. H. and Souganidis, P. E. (1986). Asymptotic series and the method of vanishing viscosity. Indiana Univ. Math. J. 35 425-447, 925.Google Scholar
Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York.Google Scholar
Glasserman, P. and Kou, S. G. (1995). Analysis of an importance sampling estimator for tandem queues. ACM Trans. Model. Comput. Simul. 5 22-42.Google Scholar
Glasserman, P. and Wang, Y. (1997). Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Prob. 7 731-746.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S-E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Vanden-Eijnden, E. and Weare, J. (2012). Rare event simulation of small noise diffusions. Commun. Pure Appl. Math. 65 1770-1803.Google Scholar