Hostname: page-component-7dd5485656-hw7sx Total loading time: 0 Render date: 2025-10-31T00:56:47.071Z Has data issue: false hasContentIssue false
  • English
  • Français

On Large Deviations for Small Noise Itô Processes

Part of: Stochastic analysis Limit theorems Functional-differential and differential-difference equations Markov processes

Published online by Cambridge University Press:  22 February 2016

Alberto Chiarini  and
Markus Fischer
Alberto Chiarini*
Affiliation:
Technische Universität Berlin
Markus Fischer*
Affiliation:
Università di Padova
*
∗ Postal address: Department of Mathematics, Technische Universität Berlin, Straβe des 17. Juni 136, 10623 Berlin, Germany. Email address: chiarini@math.tu-berlin.de
∗∗ Postal address: Department of Mathematics, Università di Padova, via Trieste 63, 35121 Padova, Italy. Email address: fischer@math.unipd.it
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.

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate Itô stochastic differential equations with predictable coefficients, which may also depend on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.

Keywords

Large deviationItô processstochastic differential equationFreidlin-Wentzell estimatetime delayCIR processweak convergence

MSC classification

Primary: 60F10: Large deviations 60H10: Stochastic ordinary differential equations
Secondary: 60J60: Diffusion processes 34K50: Stochastic functional-differential equations

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 1126 - 1147
Copyright
© Applied Probability Trust 

References

Azencott, R. (1980). Grandes déviations et applications. In Eighth Saint Flour Probability Summer School—1978 (Lecture Notes Math. 774), Springer, Berlin, pp. 1–176.Google Scholar
Baldi, P. and Caramellino, L. (2011). General Freidlin-Wentzell large deviations and positive diffusions. Statist. Prob. Lett. 81, 1218–1229.CrossRefGoogle Scholar
Baldi, P. and Chaleyat-Maurel, M. (1988). An extension of Ventsel'–Freidlin estimates. In Stochastic Analysis and Related Topics (Lecture Notes Math. 1316), Springer, Berlin, pp. 305–327.CrossRefGoogle Scholar
Boué, M. and Dupuis, P. (1998). A variational representation for certain functionals of Brownian motion. Ann. Prob. 26, 1641–1659.CrossRefGoogle Scholar
Budhiraja, A. and Dupuis, P. (2000). A variational representation for positive functionals of infinite dimensional Brownian motion. Prob. Math. Statist. 20, 39–61.Google Scholar
Budhiraja, A., Dupuis, P. and Maroulas, V. (2008). Large deviations for infinite dimensional stochastic dynamical systems. Ann. Prob. 36, 1390–1420.CrossRefGoogle Scholar
Budhiraja, A., Dupuis, P. and Maroulas, V. (2011). Variational representations for continuous time processes. Ann. Inst. H. Poincaré Prob. Statist. 47, 725–747.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Donati-Martin, C., Rouault, A., Yor, M. and Zani, M. (2004). Large deviations for squares of Bessel and Ornstein–Uhlenbeck processes. Prob. Theory Relat. Fields 129, 261–289.CrossRefGoogle Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.CrossRefGoogle Scholar
Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (Graduate Texts Math. 113), 2nd edn. Springer, New York.Google Scholar
Mohammed, S-E. A. and Zhang, T. (2006). Large deviations for stochastic systems with memory. Discrete Contin. Dyn. Syst. B 6, 881–893.Google Scholar
Priouret, P. (1982). Remarques sur les petites perturbations de systèmes dynamiques. In Seminar on Probability, XVI (Lecture Notes Math. 920), Springer, Berlin, pp. 184–200.Google Scholar
Puhalskii, A. A. (2001). Large Deviations and Idempotent Probability (Monogr. Surveys Pure Appl. Math. 119). Chapman & Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Puhalskii, A. A. (2004). On some degenerate large deviation problems. Electron. J. Prob. 9, 862–886.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus. Cambridge University Press.Google Scholar
Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167.Google Scholar