Multiple fractional integral with Hurst parameter less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mstyle displaystyle="false"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mstyle></mml:math>

This article is devoted to define and solve an evolution equation of the form dy t = y t dt + d X t (y t ), where stands for the Laplace operator on a space of the form L p (R n ), and X is a finite dimensional noisy nonlinearity whose typical form is given by) is a γ -Hölder function generating a rough path and each f i is a smooth enough function defined on L p (R n ). The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.

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Non-linear rough heat equations

Deya

Gubinelli

Tindel

2011

Probab. Theory Relat. Fields

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On the lack of Gaussian tail for rough line integrals along fractional Brownian paths

Boedihardjo,

Geng

2023

Probab. Theory Relat. Fields

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We show that the tail probability of the rough line integral $$\int _{0}^{1}\phi (X_{t})dY_{t}$$ ∫ 0 1 ϕ ( X t ) d Y t , where (X, Y) is a 2D fractional Brownian motion with Hurst parameter $$H\in (1/4,1/2)$$ H ∈ ( 1 / 4 , 1 / 2 ) and $$\phi $$ ϕ is a $$C_{b}^{\infty }$$ C b ∞ -function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $$\gamma $$ γ -Weibull tail with any exponent $$\gamma >2H+1$$ γ > 2 H + 1 . In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $$C_{b}^{\infty }$$ C b ∞ . This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp.

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Group sequential tests under fractional Brownian motion in monitoring clinical trials

Lai

2010

Stat Methods Appl

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Multiple fractional integral with Hurst parameter less than 12

Cited by 20 publications

References 18 publications

Non-linear rough heat equations

Non-linear rough heat equations

On the lack of Gaussian tail for rough line integrals along fractional Brownian paths

Group sequential tests under fractional Brownian motion in monitoring clinical trials

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