The SIML method without microstructure noise

The SIML (abbreviation of Separating Information Maximal Likelihood) method, has been introduced by N. Kunitomo and S. Sato and their collaborators to estimate the integrated volatility of high-frequency data that is assumed to be an Itô process but with so-called microstructure noise. The SIML estimator turned out to share many properties with the estimator introduced by P. Malliavin and M. E. Mancino. The present paper establishes the consistency and the asymptotic normality under a general sampling scheme but without microstructure noise. Specifically, a fast convergence shown for Malliavin–Mancino estimator by E. Clement and A. Gloter is also established for the SIML estimator.

Data sharing is not applicable to the present paper as no new data were created or analyzed in this study.

1. It has been pointed out that the “bias” \( \textbf{E} [ V^{j,j'} - \widehat{\Sigma ^{j,j'}_{n,m_n}}(0)] \) converges to zero and the mean square error \( \textbf{E} [ (V^{j,j'} - \widehat{\Sigma ^{j,j'}_{n,m_n}}(q) )^2 \) does not diverge when \( m_n = o (n) \) as \( n \rightarrow \infty \), which is not the case with the realized volatility.

2. To be precise, they used the IBP technique to prove the convergence of \( \textrm{Res}_t \) in (3.1) and \( \langle M, W\rangle \), but used another approach to prove the convergence of \( I^2 \) for which the Malliavin differentiability for b is not required.

3. Here we avoid using the commonly used notation D for the derivative so as not to mix it up with the Dirichlet kernel.

4. As is remarked in Clément and Gloter (2011), the conditions (3.10) and (3.11) are not too strong. For example, the solution for a stochastic differential equation with smooth bounded coefficients naturally satisfies these conditions.

5. Misaki and Kunitomo (2015) studied a random sampling scheme for SIML in a mild situation.

Authors and Affiliations

1. Department of Mathematical Science, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan

   Jirô Akahori

2. Department of Mathematical Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555, Japan

   Ryuya Namba

3. Graduate School of Science and Engineering, Ritsumeikan University, 1-1-1, Noji-Higashi, Kusatsu, Shiga, 525-8577, Japan

   Atsuhito Watanabe

Corresponding author

Correspondence to Atsuhito Watanabe.

Conflict of interest

The authors declare that they have no Conflict of interest.

Publisher's Note

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A Appendices

1.1 A.1 A proof of Lemma 3.3

Let us put

By extending \( \varphi ^j \) and \( \varphi ^{j'} \) periodically, \( \mathcal {D}^{j,j'}_m (u,s) = D_m(\varphi ^j(u) + \varphi ^{j'} (s)) + D_m(\varphi ^j(u) - \varphi ^{j'}(s)) \) is periodic in both u and s with the period 2. Therefore, we have

Hence, it is sufficient to show that

We put \(a:=\limsup _{m_n, n \rightarrow \infty } m_n \rho _n \) which is in \( [ 0, \infty ) \) by assumption. Moreover, let

and

For \(m_n\) and n large enough, we see that

Therefore, we obtain

Since it holds that

for \( x \in \textbf{R} \), it follows from (A.3) that

Since (A.2) implies, for sufficiently large n, \(|u-c|>\frac{2a+2}{m_n} \Rightarrow \left| \varphi ^{j}(u)-c\right| \ge |u-c|-\frac{2a}{m_n}\), one has

(by changing variables with \( w = \frac{m_n}{2} |u-c|-a\))

This establishes (A.1) since clearly one has

by (A.4). \(\square \)

1.2 A.2 A proof of Proposition 3.1

Under the condition that \( \rho _n m_n^2 \rightarrow 0 \), we have

by a similar argument as the one we did for the proof of Lemma 2.1. Therefore, it suffices to prove that

We note that

holds by extending f(s, u) from \( \{ (s,u): s\le u \} \) to [0, 1] symmetrically. Then, by letting

we have

On the other hand, it follows from the expression (2.9) that

Since it holds that

we have

Then, by Lemma 3.3, we see that

Finally we shall prove the convergence of the first term in (A.7). Recalling (2.5), we have

We rely on the uniformly continuity of f. For arbitrary sufficiently small \( \varepsilon > 0 \), we can take \( \delta > 0 \) such that

Let

and

Then clearly \( (s,u) \in A^{\pm }_\delta \) satisfies \( |s-u|< \delta \) and therefore \( |f(s,u)-f(s,s)| < \varepsilon \). Then, we can bound the right-hand-side of (A.8) by

Since

and

we have

which shows the convergence to zero (as \( m \rightarrow \infty \)) of the first term in (A.7). \(\square \)

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