### Abstract

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].

Original language | English |
---|---|

Pages (from-to) | 270-322 |

Number of pages | 53 |

Journal | Journal of Functional Analysis |

Volume | 243 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Feb 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymptotic expansions
- Itô functional
- Laplace approximation
- Large deviation principle
- Rough path theory
- Stochastic Taylor expansion

### ASJC Scopus subject areas

- Analysis

### Cite this

**Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths.** / Inahama, Yuzuru; Kawabi, Hiroshi.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 243, no. 1, pp. 270-322. https://doi.org/10.1016/j.jfa.2006.09.016

}

TY - JOUR

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

AU - Inahama, Yuzuru

AU - Kawabi, Hiroshi

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

UR - http://www.scopus.com/inward/record.url?scp=33846174638&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846174638&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2006.09.016

DO - 10.1016/j.jfa.2006.09.016

M3 - Article

VL - 243

SP - 270

EP - 322

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -