[Paper Review] Asymptotic expansion of a variation with anticipative weights

Nakahiro Yoshida|arXiv (Cornell University)|Dec 31, 2020

Stochastic processes and financial applications74 references4 citations

TL;DR

This paper develops an asymptotic expansion for a variation of a Wiener process with anticipative (non-predictable) weights using Malliavin calculus and Skorohod integral theory. It introduces a novel exponent-based classification of Wiener functionals to systematically analyze stochastic expansions, leading to an expansion formula involving quasi-torsion, quasi-tangent, and other random symbols, with applications to robust volatility estimation in high-frequency finance.

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 ${\cal L}$ of certain sequences $({\cal I}_n)_{n\in{\mathbb N}}$ of Wiener functionals and we give a systematic way of estimation of the order of $({\cal I}_n)_{n\in{\mathbb N}}$. Based on this method, we introduce a notion of exponent of the sequence $({\cal I}_n)_{n\in{\mathbb N}}$, and investigate the stability and contraction effect of the operators $D_{u_n}$ and $D$ on ${\cal L}$, 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.

Motivation & Objective

• To derive an asymptotic expansion for a variation of a Wiener process with anticipative weights, which are not adapted to the Brownian filtration.
• To develop a systematic framework for estimating the order of stochastic expansions involving sequences of Wiener functionals.
• To introduce and analyze the concept of an 'exponent' for sequences of Wiener functionals to classify the contribution of each term in the expansion.
• To establish stability and contraction properties of Malliavin derivatives and divergence operators on a newly defined class $\mathcal{L}$ of functionals.
• To apply the resulting expansion to statistical problems, particularly robust volatility estimation in high-frequency data.

Proposed method

• The paper uses the theory of asymptotic expansion for Skorohod integrals with mixed normal limits as the foundational framework.
• It defines a class $\mathcal{L}$ of sequences $({\cal I}_n)_{n\in\mathbb{N}}$ of Wiener functionals to analyze the stochastic expansion of the variation variable.
• A new notion of 'exponent' is introduced to classify the order of magnitude of each term in the expansion, based on the structure of multiple Wiener integrals and their derivatives.
• The method involves estimating the $L^p$-norms of multiple Wiener integrals and their derivatives using product formulas and moment estimates.
• The stability and contraction effects of the Malliavin derivative $D$ and the divergence operator $\delta$ on $\mathcal{L}$ are analyzed via a recursive structure of the expansion.
• The expansion is constructed through a martingale-type decomposition and bounds on the derivatives of the quasi-torsion and quasi-tangent symbols.

Experimental results

Research questions

• RQ1 How can one systematically classify and estimate the order of terms in the stochastic expansion of a variation with anticipative weights?
• RQ2 What is the role of the Malliavin derivative and divergence operator in controlling the asymptotic behavior of such expansions?
• RQ3 How does the exponent of a sequence of Wiener functionals determine the leading-order contribution in the expansion?
• RQ4 What conditions ensure the stability and contraction of the operators $D_{u_n}$ and $D$ on the class $\mathcal{L}$?
• RQ5 Can the derived expansion be applied to improve robust volatility estimation in high-frequency financial data?

Key findings

• The asymptotic expansion of the variation $V_n$ is derived in terms of quasi-torsion, quasi-tangent, and other random symbols, providing a refined approximation beyond the central limit theorem.
• The exponent of a sequence $({\cal I}_n)$ is defined and used to classify the contribution of each term, enabling precise order estimation in the expansion.
• The $L^p$-norm of the remainder term in the expansion is shown to be $O(n^{-1})$ under suitable conditions, improving upon previous bounds.
• For a polynomial of multiple Wiener integrals, the $L^{2k}$-norm is bounded by $C n^{-\frac{1}{2}\xi}$ with $\xi = \mathbf{p} \cdot \mathbf{q} - m - \#\{i: p_i \geq 2 \text{ and } p_i q_i \text{ even}\}$, establishing a sharp decay rate.
• The derivative of the quasi-torsion satisfies $\|D_{u_n}D_{u_n}D_{u_n}M_n\|_p = o(r_n)$, which is crucial for the martingale expansion approach.
• An application to robust volatility estimation is outlined, showing the expansion's relevance in statistical inference for diffusion processes.