Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

Ouahra, M. Ait; Sghir, Aissa

doi:10.1080/07362994.2017.1309979

Cited by 3 publications

(9 citation statements)

References 23 publications

(35 reference statements)

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“…Note that the integrand clearly converges to 0 pointwise on T j (t) × R j . Indeed, the constant c H (see (1)) defines a continuous function of H ∈ (0, 1). Now, we proceed to bound the integrand by an integrable function.…”

Section: Identification Of the Limitmentioning

confidence: 99%

“…In [15,26], the same kind of continuity property has been tackled for the local time of the fractional Brownian motion and other Gaussian fields. Eventually, in the recent paper [1], the continuity property has been shown for additive functionals of the sub-fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

“…This result has already been proved in [4,Thm. 4.2] for the case H < 1 2 , and we extend it to H ≥ 1 2 . We point out that the latter case, in which the noise is more regular, entails some extra difficulties due to the fact that the underlying Hilbert space associated to the noise's covariance contains distributions.…”

Section: Introductionmentioning

confidence: 99%

“…The above strategy will be made clearer in Section 4 below, but let us remark at this point that the main strategy in this part of the paper does not require a separate analysis for the cases H < 1 2 and H ≥ 1 2 . Furthermore, the methodology used in both results on tightness and the limit identification cover equations (SWE) and (SHE) at the same time.…”

Section: Introductionmentioning

confidence: 99%

“…1], where the author considers a general non-linear coefficient σ(u H (t, x)) in front of the noise. We believe that such diffusion coefficient could be also considered in the case of the wave equation with H > 1 2 , but we have chosen to stick to the linear multiplicative noise in order to find a unified result that covers also the case H < 1 2 , which is more mathematically demanding.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

Giordano¹,

María²,

Quer-Sardanyons³

2020

Bernoulli

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In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H ∈ ( 1 4 , 1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law. MSC 2010: 60B10; 60H07; 60H15At this point, we aim to extend the random fieldW H defined in (9) to an isonormal Gaussian process in H H . We need the following corollary of [5, Thm. 4.3]:Proposition 2.6. The space of finite linear combinations of functions of the form f (r, z) = 1 (s,t]×(x,y] (r, z), with 0 ≤ s < t and x < y, is dense in the Hilbert space H H .Proof. The result is a direct consequence of [5, Thm. 4.3]. Indeed, in the latter paper it is proved that any predictable process {X(t, x), (t, x) ∈ R + × R} belonging to L 2 (Ω; H H ) can be approximated by finite linear combinations of processes of the form (r, z, ω) → 1 G (ω)1 (s,t] (r)1 (x,y] (z), for some G ∈ F. To prove our result, it suffices to observe that, if we choose a deterministic element ϕ in their proof, also its approximating sequence ϕ n is deterministic, and the norm in the space L 2 (Ω; H H ) coincides with the norm in H H for deterministic elements.

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Section: Identification Of the Limitmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

Giordano¹,

María²,

Quer-Sardanyons³

2020

Bernoulli

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SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

Giordano

María

Quer-Sardanyons

2020

Stochastic Processes and their Applications

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On the existence and the Hölder regularity of the local time of the Brownian bridge

Allaoui

Sghir

Hadiri

2022

Random Operators and Stochastic Equations

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In this paper, we will establish the existence and the Hölder regularity of the local time of the Brownian bridge. Our results are obtained by using a result on Malliavin calculus in [K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor, Occupation densities for certain processes related to fractional Brownian motion, Stochastics 82 2010, 1–3, 133–147] for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach based on the concept of local nondeterminism for Gaussian processes introduced by Berman [S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 1973/74, 69–94].

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Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

Cited by 3 publications

References 23 publications

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

SPDEs with fractional noise in space: Continuity in law with respect to the Hurst index

SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

On the existence and the Hölder regularity of the local time of the Brownian bridge

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