Forward, backward and symmetric stochastic integration

Russo, Francesco; Vallois, Pierre

doi:10.1007/bf01195073

Cited by 260 publications

(220 citation statements)

References 16 publications

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“…In the context of uniform convergence in probability the latter corresponds to a slight extension of the so-called forward integral of Russo and Vallois [6] for stochastic processes. More details and relationships to other integrals may be found in [14].…”

Section: Dx(t) = a 0 X(t) T Dw (T) + M J=1 A J X(t) T Db Hj (T) + Bmentioning

confidence: 99%

Stochastic differential equations with fractal noise

Zähle

2005

Mathematische Nachrichten

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Key wordsStochastic differential equations in R n with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W β 2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W α 2 for 0 < α < 1. Linear equations are considered as a special case.This approach leads to fast computer algorithms basing on Picard's iteration method.

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Section: Dx(t) = a 0 X(t) T Dw (T) + M J=1 A J X(t) T Db Hj (T) + Bmentioning

confidence: 99%

Stochastic differential equations with fractal noise

Zähle

2005

Mathematische Nachrichten

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“…One needs generalized alternative ways to integrate stochastically with respect to such processes. In general these generalized method are essentially of two types: the first is the pathwise type calculus and (here we included the rough path analysis [35] and the stochastic calculus via regularization [37]) and the second type is Malliavin calculus and the Skorohod integration theory [29]. In general the pathwise type calculus is connected to the trajectorial regularity and/or the covariance structure of the integrator process.…”

Section: Pathwise Stochastic Calculusmentioning

confidence: 99%

“…We define in Section 4 the Hilbert-valued Rosenblatt process and we consider stochastic evolution equations with this process as noise. Section 5 describes the application of the stochastic calculus via regularization introduced by Russo and Vallois in [37] to the Rosenblatt process and in Section 6 we discuss the Skorohod (divergence) integral: we define the integral and we give conditions that ensure the integrability and the continuity of the indefinite integral process. In Section 7 we prove the relation between the pathwise and the divergence integrals: here the pathwise integral is equal to the Skorohod integral plus two trace terms (in the fBm case there is only a trace term).…”

Section: Introductionmentioning

confidence: 99%

“…The first approach (that includes essentially the rough paths analysis, see [35], and the stochastic calculus via regularization, see [37]) can be directly applied to the Rosenblatt process because of its regular paths and of the nice covariance structure; a pathwise Itô formula can be written and Stratonovich stochastic equation with Z as noise can be considered. The Malliavin calculus and the Skorohod integration are in general connected in a deeper way to the Gaussian structure of the integrator process and as it will be seen in the present work, Skorohod Itô formula can be derived in a complete form only in particular cases.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the Rosenblatt process

Tudor

2008

ESAIM: PS

155

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Abstract. We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.Mathematics Subject Classification. 60G12, 60G15, 60H05, 60H07.

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“…León and Tudor [10]) work with the stochastic Stratonovich integral in the Russo and Vallois sense [18] when H ∈ (1/4, 1/2) (resp. H ∈ (1/2, 1)).…”

Section: Introductionmentioning

confidence: 99%

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

León

Martı́n

2007

Stochastic Analysis and Applications

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In this paper we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here the coefficients are deterministic, the inital condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results. IntroductionThe fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1) is a Gaussian process with useful properties. In particular, the stationary of its increments, and the self-similarity and the long-range dependence of this process (see Mandelbrot and Van Ness [13]) become the fBm a suitable driven noise for the construction of stochastic models and the analysis of phenomena that exhibit scale-invariant and long-range correlated force. However the fBm is not a semimartingale when H = 1 2 . Hence we cannot apply the techniques of the stochastic calculus in the Itô sense to define a stochastic integral with respect to the fBm.Different interpretations of stochastic integral with respect to the fBm B have been used by several authors to study the fractional stochastic differential equation of the form(1.1)In the case that H ∈ (1/2, 1), it is reasonable to consider equation (1.1) as a path-by-path ordinary differential equation since B has Hölder-continuous paths with all exponents less than H and T 0 Y s dB s exists as a pathwise Riemann-Stieltjes integral for any λ-Hölder continuous process Y with λ > 1 − H (see Young [22]). This pathwise equation has been studied by several authors (see for instance [7, 8, 11, 19] and [23]). Zähle [23] has improved this pathwise approach * Partially supported by CONACyT

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Forward, backward and symmetric stochastic integration

Cited by 260 publications

References 16 publications

Stochastic differential equations with fractal noise

Stochastic differential equations with fractal noise

Analysis of the Rosenblatt process

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

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