Order estimate of functionals related to fractional Brownian motion

Nualart and Yoshida (2019) presented a general scheme for asymptotic expansion of Skorohod integrals. However, when applying the general theory to a variation of a process related to a fractional Brownian motion, one repeatedly needs estimates of the order of Lp-norms and Sobolev norms of functionals that are complicated as a randomly weighted sum of products of multiple integrals of the fractional Brownian motion. To resolve the difficulties, in this paper, we construct a theory of exponents based on a graphical representation of the structure of the functionals, which enables us to tell an upper bound of the order of functionals by a simple rule. We also show how the exponents change by the actions of the Malliavin derivative and its projection. The exponents are useful in various studies of limit theorems as well as in applications to asymptotic expansion. (c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).