TY - JOUR

T1 - Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths

AU - Inahama, Yuzuru

AU - Kawabi, Hiroshi

N1 - Funding Information:
The authors express their deepest gratitudes to Professor Shigeki Aida for valuable suggestions. They also thank Professors Sergio Albeverio, Bruce Driver, David Elworthy, Shizan Fang, Paul Malliavin, Zhongmin Qian and Shinzo Watanabe for their helpful comments and encouragements. The authors were supported by JSPS Research Fellowships for Young Scientists and the second author is supported by 21st century COE program “Development of Dynamic Mathematics with High Functionality” at Faculty of Mathematics, Kyushu University.

PY - 2007/2/1

Y1 - 2007/2/1

N2 - In this paper, we establish asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace's method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148-194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797-816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach space-valued Itô functionals of Brownian rough paths, in: Proceedings of the Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Itô, Springer, Berlin, in press. Available at: http://www.abelprisen.no/no/abelprisen/deltagere_2005.html].

AB - In this paper, we establish asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths under the condition that the phase function has finitely many non-degenerate minima. Our main tool is the Banach space-valued rough path theory of T. Lyons. We use a large deviation principle and the stochastic Taylor expansion with respect to the topology of the space of geometric rough paths. This is a continuation of a series of papers by Inahama [Y. Inahama, Laplace's method for the laws of heat processes on loop spaces, J. Funct. Anal. 232 (2006) 148-194] and by Inahama and Kawabi [Y. Inahama, H. Kawabi, Large deviations for heat kernel measures on loop spaces via rough paths, J. London Math. Soc. 73 (3) (2006) 797-816], [Y. Inahama, H. Kawabi, On asymptotics of certain Banach space-valued Itô functionals of Brownian rough paths, in: Proceedings of the Abel Symposium 2005, Stochastic Analysis and Applications, A Symposium in Honor of Kiyosi Itô, Springer, Berlin, in press. Available at: http://www.abelprisen.no/no/abelprisen/deltagere_2005.html].

KW - Asymptotic expansions

KW - Itô functional

KW - Laplace approximation

KW - Large deviation principle

KW - Rough path theory

KW - Stochastic Taylor expansion

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U2 - 10.1016/j.jfa.2006.09.016

DO - 10.1016/j.jfa.2006.09.016

M3 - Article

AN - SCOPUS:33846174638

VL - 243

SP - 270

EP - 322

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -