Order estimate of functionals related to fractional Brownian motion and asymptotic expansion of the quadratic variation of fractional stochastic differential equation

2022/06/01 by Hayate Yamagishi, Yamagishi, Hayate, Nakahiro Yoshida +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2206.00323

openalex publication_date 2022/06/01 · openalex created_date 2023/02/14 · openalex updated_date 2026/07/28

Abstract

We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we introduce two types of exponents by means of the "weighted graphs" capturing the structure of the sum in the functional, and investigate how the exponents change by the action of the Malliavin derivative and its projection.

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