On Asymptotics of Banach Space-valued Itô Functionals of Brownian Rough Paths

Abstract

In this paper, we discuss asymptotics for certain Banach spacevalued Itô functionals of Brownian rough paths based on the results of Inahama-Kawabi [10] and Inahama [9]. Our main tool is the Banach spacevalued rough path theory of T. Lyons. As examples, we deal with heat processes on loop spaces and solutions of the stochastic differential equations (SDEs) on M-type 2 Banach spaces.

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

“…In the following, we discuss the higher integrability of exp(−J 2 F (φ)(·)) based on [7] and [9]. First we give an explicit representation of the Hessian A which are defined in Section 3.…”

“…For heat processes on loop spaces, Laplace's method was studied in the earlier paper Inahama [7]. In this paper, as a continuation of [7] and [9], we establish the asymptotic expansion formulas for wider classes of (infinitedimensional) Banach space-valued Wiener functionals by using the fact that the rough path theory of T. Lyons works on any Banach space.…”

“…As examples of (1.1), we can give a class of heat processes on loop spaces and the solutions of SDEs on M-type 2 Banach spaces. See Inahama and Kawabi [9] for details.…”

Sensitivity of rough differential equations: An approach through the Omega lemma

“…The map I(f, g, b, y 0 , y † 0 ) giving the solution to (23) is then of class C k+(1−κ) min{δ,γ} . Using for z the decomposition z † = 1 and z = 0, and replacing the vector field f by f , it is easily seen that we may consider the problem Asymptotic expansions in can then be performed as in [5,[32][33][34][35].…”

Rough integrators on Banach manifolds

“…The interest of working with Banach manifolds comes from the fact that they naturally pop up in a number of geometric situations, as path or loop spaces over some finite dimensional manifold, as in the works [7,8] of Brzezniak, Carroll and Elworthy, or the works [9,10] of Inahama and Kawabi, or as manifolds of maps of a given finite dimensional manifold, as in the works [11,12] of Elworthy and Brzezniak, to mention but a few works from the probability community. Note however that many interesting infinite dimensional manifolds are Fréchet manifolds, for which no theory of rough paths is presently available.…”