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
SN - 0022-1236
VL - 243
SP - 270
EP - 322
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -