Tanaka formula for the fractional Brownian motion

Coutin, Laure; Nualart, David; Tudor, Ciprian A.

doi:10.1016/s0304-4149(01)00085-0

Cited by 56 publications

(62 citation statements)

References 15 publications

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“…The main arguments we follow can be considered classical, and are found for example in [8]. Let us denote by p (x) = (2 ) −1/2 exp(−x 2 /(2 )) the heat kernel.…”

Section: Chaos Decompositionmentioning

confidence: 99%

“…This was done for example originally in Berman's paper [4]. On the other hand, and more recently, several stochastic analysts working on fractional Brownian motion have chosen to consider a different occupation measure because it yields a connection to stochastic calculus via the Itô-Tanaka formula: see for example [8]; also see the summary on local time for fBm-based processes in [19].…”

Section: Introductionmentioning

confidence: 99%

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Skorohod integration and stochastic calculus beyond the fractional Brownian scale

Mocioalca

Viens

2005

Journal of Functional Analysis

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We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.

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“…The main arguments we follow can be considered classical, and are found for example in [8]. Let us denote by p (x) = (2 ) −1/2 exp(−x 2 /(2 )) the heat kernel.…”

Section: Chaos Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

Mocioalca

Viens

2005

Journal of Functional Analysis

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“…The canonical example of such processes is the fractional Brownian sheet. It was studied in [16] for H > 1/2, where Itô and Tanaka formulas were established, the former formula being the canonical chain rule of stochastic calculus, its cornerstone, and the latter being a representation of fBm's local time (occupation time density) using a stochastic integral (see [5] for the results with only one parameter). The purpose of this article is to show that the techniques of [16], which only apply to the case of H > 1/2, can be supplanted by developing a stochastic integration that works also for the fractional Brownian sheet with any Hurst parameters less than 1/2, and beyond the fractional scale, using the ideas of [11].…”

Section: Introductionmentioning

confidence: 99%

Itô formula for the two-parameter fractional Brownian motion using the extended divergence operator

Tudor

Viens

2006

Stochastics

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We develop a stochastic calculus of divergence type with respect to the fractional Brownian sheet (fBs) with any Hurst parameters in (0, 1) and beyond the fractional scale. We define stochastic integration in the extended Skorohod sense, and derive Itô and Tanaka formulas. In the case of Gaussian fields that are more irregular than fBs for any Hurst parameters, we are able to complete the same program for those Gaussian fields that are almost-surely uniformly continuous.2000 Mathematics Subject Classification: 60H05, 60G15, 60G18.

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“…The classical idea of approximating the Dirac distribution δ x by p ε has been used to calculate the chaotic decomposition of the local time in the case of the Brownian motion by Nualart and Vives [11] and for the fractional Brownian motion by Coutin et al [3] and Eddahbi et al [4]. Before stating precise results of this section, we prove some technical lemmas.…”

Section: Eddahbi Et Al 219mentioning

confidence: 98%

A Stroock formula for a certain class of Lévy processes and applications to finance

Eddahbi

Solé

Vives

2005

International Journal of Stochastic Analysis

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We find a Stroock formula in the setting of generalized chaos expansion introduced by Nualart and Schoutens for a certain class of Lévy processes, using a Malliavin-type derivative based on the chaotic approach. As applications, we get the chaotic decomposition of the local time of a simple Lévy process as well as the chaotic expansion of the price of a financial asset and of the price of a European call option. We also study the behavior of the tracking error in the discrete delta neutral hedging under both the equivalent martingale measure and the historical probability.

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Tanaka formula for the fractional Brownian motion

Cited by 56 publications

References 15 publications

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

Itô formula for the two-parameter fractional Brownian motion using the extended divergence operator

A Stroock formula for a certain class of Lévy processes and applications to finance

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