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George Tauchen - One of the best experts on this subject based on the ideXlab platform.
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limit theorems for the empirical distribution function of scaled increments of ito Semimartingales at high frequencies
Annals of Applied Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of “devolatilized” increments of an Ito semimartingale observed at high frequencies. These “ devolatilized” increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and “large” jumps. We derive the limit of the empirical cdf of the adjusted increments for any Ito semimartingale whose dominant component at high frequencies has activity index of 1 < β ≤ 2, where β = 2 corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Ito Semimartingales with non-vanishing diffusion coefficient against Ito Semimartingales with no diffusion component.
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limit theorems for the empirical distribution function of scaled increments of ito Semimartingales at high frequencies
Annals of Applied Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of “devolatilized” increments of an Ito semimartingale observed at high frequencies. These “devolatilized” increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and “large” jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Ito semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Ito Semimartingales with nonvanishing diffusion coefficient against Ito Semimartingales with no diffusion component.
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limit theorems for the empirical distribution function of scaled increments of it o Semimartingales at high frequencies
arXiv: Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of "devolatilized" increments of an It\^{o} semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any It\^{o} semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of It\^{o} Semimartingales with nonvanishing diffusion coefficient against It\^{o} Semimartingales with no diffusion component.
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Limit theorems for power variations of pure-jump processes with application to activity estimation
arXiv: Probability, 2011Co-Authors: Viktor Todorov, George TauchenAbstract:This paper derives the asymptotic behavior of realized power variation of pure-jump It\^{o} Semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled It\^{o} semimartingale over a fixed interval.
Andreas Basse - One of the best experts on this subject based on the ideXlab platform.
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spectral representation of gaussian Semimartingales
Journal of Theoretical Probability, 2009Co-Authors: Andreas BasseAbstract:The aim of the present paper is to characterize the spectral representation of Gaussian Semimartingales. That is, we provide necessary and sufficient conditions on the kernel K for Xt=∫Kt(s) dNs to be a semimartingale. Here, N denotes an independently scattered Gaussian random measure on a general space S. We study the semimartingale property of X in three different filtrations. First, the ℱX-semimartingale property is considered, and afterwards the ℱX,∞-semimartingale property is treated in the case where X is a moving average process and ℱtX,∞=σ(Xs:s∈(−∞,t]). Finally, we study a generalization of Gaussian Volterra processes. In particular, we provide necessary and sufficient conditions on K for the Gaussian Volterra process ∫−∞tKt(s) dWs to be an ℱW,∞-semimartingale (W denotes a Wiener process). Hereby we generalize a result of Knight (Foundations of the Prediction Process, 1992) to the nonstationary case.
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levy driven moving averages and Semimartingales
Stochastic Processes and their Applications, 2009Co-Authors: Andreas Basse, Jan Skov PedersenAbstract:The aim of the present paper is to study the semimartingale property of continuous time moving averages driven by Levy processes. We provide necessary and sufficient conditions on the kernel for the moving average to be a semimartingale in the natural filtration of the Levy process, and when this is the case we also provide a useful representation. Assuming that the driving Levy process is of unbounded variation, we show that the moving average is a semimartingale if and only if the kernel is absolutely continuous with a density satisfying an integrability condition.
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gaussian moving averages and Semimartingales
Electronic Journal of Probability, 2008Co-Authors: Andreas BasseAbstract:In the present paper we study moving averages (also known as stochastic convolutions) driven by a Wiener process and with a deterministic kernel. Necessary and sufficient conditions on the kernel are provided for the moving average to be a semimartingale in its natural filtration. Our results are constructive - meaning that they provide a simple method to obtain kernels for which the moving average is a semimartingale or a Wiener process. Several examples are considered. In the last part of the paper we study general Gaussian processes with stationary increments. We provide necessary and sufficient conditions on spectral measure for the process to be a semimartingale.
Viktor Todorov - One of the best experts on this subject based on the ideXlab platform.
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limit theorems for the empirical distribution function of scaled increments of ito Semimartingales at high frequencies
Annals of Applied Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of “devolatilized” increments of an Ito semimartingale observed at high frequencies. These “ devolatilized” increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and “large” jumps. We derive the limit of the empirical cdf of the adjusted increments for any Ito semimartingale whose dominant component at high frequencies has activity index of 1 < β ≤ 2, where β = 2 corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Ito Semimartingales with non-vanishing diffusion coefficient against Ito Semimartingales with no diffusion component.
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limit theorems for the empirical distribution function of scaled increments of ito Semimartingales at high frequencies
Annals of Applied Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of “devolatilized” increments of an Ito semimartingale observed at high frequencies. These “devolatilized” increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and “large” jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Ito semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of Ito Semimartingales with nonvanishing diffusion coefficient against Ito Semimartingales with no diffusion component.
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limit theorems for the empirical distribution function of scaled increments of it o Semimartingales at high frequencies
arXiv: Probability, 2014Co-Authors: Viktor Todorov, George TauchenAbstract:We derive limit theorems for the empirical distribution function of "devolatilized" increments of an It\^{o} semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any It\^{o} semimartingale whose dominant component at high frequencies has activity index of $1<\beta\le2$, where $\beta=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use the developed limit theory to construct a feasible and pivotal test for the class of It\^{o} Semimartingales with nonvanishing diffusion coefficient against It\^{o} Semimartingales with no diffusion component.
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Limit theorems for power variations of pure-jump processes with application to activity estimation
arXiv: Probability, 2011Co-Authors: Viktor Todorov, George TauchenAbstract:This paper derives the asymptotic behavior of realized power variation of pure-jump It\^{o} Semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled It\^{o} semimartingale over a fixed interval.
Francesco Russo - One of the best experts on this subject based on the ideXlab platform.
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Generalized covariation for Banach space valued processes, Itô formula and applications
2015Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily Semimartingales) and related Itô formula. If X and Y take respectively values in Banach spaces B1 and B2 and χ is a suitable subspace of the dual of the projective tensor product of B1 and B2 (denoted by (B1⊗̂πB2) ∗), we define the so-called χ-covariation of X and Y. If X = Y, the χ-covariation is called χ-quadratic variation. The notion of χ-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if χ is the whole space (B1⊗̂πB1) ∗ then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = C([−τ, 0]) for some τ> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−τ, 0])-valued process X: = X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0]. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type h = H(XT (·)), H: C([−T, 0]) − → R for not-necessarily Semimartingales X with finite quadratic variation. This representation will be linked to a function u: [0, T]×C([−T, 0]) − → R solving an infinite dimensional partial differential equation
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generalized covariation for banach space valued processes ito formula and applications
Osaka Journal of Mathematics, 2014Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily Semimartingales) and related Ito formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Metivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Ito formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily Semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
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Generalized covariation for Banach space valued processes, Itô formula and applications
2013Co-Authors: Cristina Di Girolami, Francesco RussoAbstract:This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily Semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily Semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
Thomas Delerue - One of the best experts on this subject based on the ideXlab platform.
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mixed Semimartingales volatility estimation in the presence of fractional noise
Social Science Research Network, 2021Co-Authors: Carsten Chong, Thomas DelerueAbstract:We consider the problem of estimating volatility for high-frequency data when the observed process is the sum of a continuous Ito semimartingale and a noise process that locally behaves like fractional Brownian motion with Hurst parameter H. The resulting class of processes, which we call mixed Semimartingales, generalizes the mixed fractional Brownian motion introduced by Cheridito [Bernoulli 7 (2001) 913–934] to time-dependent and stochastic volatility. Based on central limit theorems for variation functionals, we derive consistent estimators and asymptotic confidence intervals for H and the integrated volatilities of both the semimartingale and the noise part, in all cases where these quantities are identifiable. When applied to recent stock price data, we find strong empirical evidence for the presence of fractional noise, with Hurst parameters H that vary considerably over time and between assets.
