Edgeworth expansion for functionals of continuous diffusion processes

Podolskij, Mark; Yoshida, Nakahiro

doi:10.1214/16-aap1179

Cited by 16 publications

(14 citation statements)

References 31 publications

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“…for F = 1 and G = G • R [24]. There are two terms R[24, i] (i = 1, 2) according to (7.28) for F = 1 and G =Ĝ (1) n (θ; z, x).…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 99%

“…There are two terms R[24, i] (i = 1, 2) according to (7.28) for F = 1 and G =Ĝ (1) n (θ; z, x). To R [24,1], useď + 4 IBP with (7.23) and (7.17). To R [24,2],ď + 2 IBP with (7.22) and (7.17).…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 99%

“…To R [24,1], useď + 4 IBP with (7.23) and (7.17). To R [24,2],ď + 2 IBP with (7.22) and (7.17). Then sup (z,x)∈Λn(ď) |(z, x)|ď +1 |\ ∂ α R[24]| = o(r n ).…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 99%

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Asymptotic expansion of Skorohod integrals

Nualart¹,

Yoshida²

2019

Electron. J. Probab.

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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.

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“…for F = 1 and G = G • R [24]. There are two terms R[24, i] (i = 1, 2) according to (7.28) for F = 1 and G =Ĝ (1) n (θ; z, x).…”

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 99%

Section: Asymptotic Expansion For Measurable Functionsmentioning

confidence: 99%

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Asymptotic expansion of Skorohod integrals

Nualart¹,

Yoshida²

2019

Electron. J. Probab.

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“…We note that this work relies on very technical tools from Malliavin calculus which are beyond the scope of this paper. Podolskij and Yoshida () apply this theory within the framework of power variations of diffusion processes. Although the last work allows leverage effects, it is assumed that σ is driven (only) by the original Brownian motion W , thereby excluding stochastic volatility models.…”

Section: Edgeworth Expansion For Realized Volatilitymentioning

confidence: 99%

Validity of Edgeworth expansions for realized volatility estimators

Hounyo

Veliyev

2016

The Econometrics Journal

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The main contribution of this paper is to establish the formal validity of Edgeworth expansions for realized volatility estimators. First, in the context of no microstructure effects, our results rigorously justify the Edgeworth expansions for realized volatility derived in Gonçalves and Meddahi (2009, Econometrica 77, 283-306). Second, we show that the validity of the Edgeworth expansions for realized volatility might not cover the optimal two-point distribution wild bootstrap proposed by Gonçalves and Meddahi. Then, we propose a new optimal nonlattice distribution, which ensures the second-order correctness of the bootstrap. Third, in the presence of microstructure noise, based on our Edgeworth expansions, we show that the new optimal choice proposed in the absence of noise is still valid in noisy data for the pre-averaged realized volatility estimator proposed by Podolskij and Vetter (2009, Bernoulli 15, 634-658). Finally, we show how confidence intervals for integrated volatility can be constructed using these Edgeworth expansions for noisy data. Our Monte Carlo simulations show that the intervals based on the Edgeworth corrections have improved the finite sample properties relatively to the conventional intervals based on the normal approximation.

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“…The Yoshida theory provides a general framework and for each of concrete applications, a nonnegligible effort is still required to obtain an explicit expression of the expansion terms. An application to the power variation of diffusion processes is given by Podolskij and Yoshida (2016). Recently, an application to the Euler-Maruyama discretization error is given by Podolskij, Veliyev, and Yoshida (2018).…”

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confidence: 99%

The asymptotic expansion of the regular discretization error of Itô integrals

Alòs

Fukasawa

2020

Mathematical Finance

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We study an Edgeworth‐type refinement of the central limit theorem for the discretization error of Itô integrals. Toward this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula. Two applications to finance are given; the asymptotics of discrete hedging error under the Black–Scholes model and the difference between continuously and discretely monitored variance swap payoffs under stochastic volatility models.

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Edgeworth expansion for functionals of continuous diffusion processes

Cited by 16 publications

References 31 publications

Asymptotic expansion of Skorohod integrals

Asymptotic expansion of Skorohod integrals

Validity of Edgeworth expansions for realized volatility estimators

The asymptotic expansion of the regular discretization error of Itô integrals

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