Asymptotic expansion of Skorohod integrals

Abstract

Asymptotic expansion of the distribution of a perturbation Z n of a Skorohod integral jointly with a reference variable X n is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation E[f (Z n , X n )] for differentiable functions f and also measurable functions f . In the latter case, the interpolation method connects the two non-degeneracies of variables for finite n and ∞. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale expansion.

Quasi-likelihood analysis and its applications

“…Relatively high order of integrability (i.e., a large p) is necessary to carry out this plot because the integration-by-parts formula in the Malliavin calculus requires algebras of variables. Asymptotic expansion of Skorokhod integrals recently presented by Nualart andYoshida (2019), combined with Yoshida (2020), is also applicable to this problem in place of the martingale expansion. The L p -boundedness of an estimator is a key to develop the asymptotic theory.…”

Adaptive estimation for degenerate diffusion processes

“…The matrix notation repeats S i s twice for the quadratic form, thrice for the cubic form, and so on. This notation was introduced by [52] and already adopted by many papers, e.g., [45], [48], [53], [31], [23], [32], [13], [37], [36], [35], just to name a few.…”

Mixed-normal limit theorems for multiple Skorohod integrals in high-dimensions, with application to realized covariance

“…for smooth functions f : R 2d → R. In fact, the remaining part of the proof is essentially the same as the one for the high-dimensional central limit theorem of [19]. To get a reasonable estimate for (A.5), we derive an interpolation formula for it, borrowing an idea from [57]. Namely, we use the duality between iterated Malliavin derivatives and multiple Skorohod integrals combined with the interpolation method in the frequency domain introduced in [57] (see also [66]).…”

“…However, this result is not directly applicable to the current situation because it assumes that the components of S † n are conditionally independent, which is less interesting to statistical applications (and especially not the case in the problem illustrated above). To remove such a restriction from the result of [53], we employ the novel interpolation method introduced in Nualart & Yoshida [57], instead of Slepian's interpolation used in [53] and the original CCK theory.…”

“…Remark 3.1. The variable ∆ n defined in (3.5) is called the quasi-tangent in[57].when ν = (ν kl ) ∈ α∈A(q j ) N * 4 (α). To see this, let us recall that D k M n , D k C n , D k W n and D k X n take values in (H ⊗k ) d , (H ⊗k ) d×d , (H ⊗k ) d and (H ⊗k ) m×d , respectively (cf.…”