Asymptotic expansion of a variation with anticipative weights

DOI10.1016/J.SPA.2022.10.003arXiv2101.00089MaRDI QIDQ6357361

Nakahiro Yoshida

Publication date: 31 December 2020

Abstract: Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and other random symbols. To specify these random symbols, it is necessary to classify the level of the effect of each term appearing in the stochastic expansion of the variable in question. To solve this problem, we consider a class $calL$ of certain sequences $({calI}_n{)}_{ninmathbbN}$ of Wiener functionals and we give a systematic way of estimation of the order of $({calI}_n{)}_{ninmathbbN}$. Based on this method, we introduce a notion of exponent of the sequence $({calI}_n{)}_{ninmathbbN}$, and investigate the stability and contraction effect of the operators $D_{u_n}$ and $D$ on $calL$, where $u_n$ is the integrand of a Skorohod integral. After constructed these machineries, we derive asymptotic expansion of the variation having anticipative weights. An application to robust volatility estimation is mentioned.

Mathematics Subject Classification ID

Stochastic integrals (60H05) Stochastic calculus of variations and the Malliavin calculus (60H07) Limit theorems in probability theory (60F99)

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