THE ROUGH PATH ASSOCIATED TO THE MULTIDIMENSIONAL ANALYTIC FBM WITH ANY HURST PARAMETER

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THE ROUGH PATH ASSOCIATED TO THE MULTIDIMENSIONAL ANALYTIC FBM WITH ANY HURST PARAMETER SAMY TINDEL AND JÉRÉMIE UNTERBERGER Abstract. In this paper, we consider a complex-valued d-dimensional fractional Brow- nian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by ?. This process has been introduced in [16], and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on ? can be constructed for any value of the Hurst parameter in (0, 1/2). This allows in par- ticular to solve differential equations driven by ? in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli [6]. 0. Introduction The (two-sided) fractional Brownian motion t ? Bt, t ? R (fBm for short) with Hurst exponent ?, ? ? (0, 1), defined as the centered Gaussian process with covariance E[BsBt] = 12(|s| 2? + |t|2? ? |t? s|2?), (1) is a natural generalization in the class of Gaussian processes of the usual Brownian motion, in the sense that it exhibits two fundamental properties shared with Brownian motion, namely, it has stationary increments, viz.

• fbm ?


• both


• when ?


• dif- ferential equations


• hölder regularity


• complex valued


• analytic fractional


• real-valued fbm


• equations driven

Sujets

• multiplication

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• Publié par: pefav
• Nombre de lectures: 18
• Langue: English

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THE ROUGH PATH ASSOCIATED TO THE MULTIDIMENSIONAL ANALYTIC FBM WITH ANY HURST PARAMETER

SAMY TINDEL AND JÉRÉMIE UNTERBERGER

Abstract.In this paper, we consider a complex-valuedd-dimensional fractional Brow- nian motion defined on the closure of the complex upper half-plane, calledanalytic fractional Brownian motionand denoted by. This process has been introduced in [16], and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on can be constructed for any value of the Hurst parameter in(0,1/2) allows in par-. This ticular to solve differential equations driven byin a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli [6].

0.Introduction

The (two-sided) fractional Brownian motiont→B t ,t∈R(fBm for short) with Hurst exponent,∈(0,1), defined as the centered Gaussian process with covariance E[B s B t 1 2 ]=(|s| 2 +|t| 2  |ts| 2 ),(1) is a natural generalization in the class of Gaussian processes of the usual Brownian motion, in the sense that it exhibits two fundamental properties shared with Brownian motion, namely, it has stationary increments, viz.E[(B t B s )(B u B v )] =E[(B t+a B s+a )(B u+a  B v+a )]for everya, s, t, u, v∈R, and it is self-similar, viz. ∀ >0,(B t , t∈R) (la = w) (  B t , t∈R).(2) One may also define ad-dimensional vector Gaussian process (called:d-dimensional frac- tional Brownian motion) by settingB t :=B t = (B t (1), . . . , B t (d)), where(B t (i), t∈ R) i=1,...,d ared theoretical interest Itsindependent (scalar) fractional Brownian motions. lies in particular in the fact that it is (up to normalization) the only Gaussian process satisfying the two properties (1) and (2). Furthermore, a standard application of Kol- mogorov’s theorem shows that fBm has a version with(²)-Hölder paths for every ² >0. This makes this process amenable to models where a Gaussian process with Hölder continuity exponent different from1/2is needed, and we refer for instance to [1, 9, 14] for some applications to biophysics. Consequently, there has been a widespread interest during the past ten years in con- structing a stochastic integration theory with respect to fBm and solving stochastic dif- ferential equations driven by fBm. The multi-dimensional case is very different from the one-dimensional case. When one tries to integrate for instance a stochastic differential equation driven by a two-dimensional fBmB= (B(1), B(2))by using any kind of Picard iteration scheme, one encounters very soon the problem of defining the Lévy area ofB

Date: October 7, 2008. 2000Mathematics Subject Classification.60H05, 60H10, 60G15. Key words and phrases.Rough paths theory; Stochastic differential equations; Fractional Brownian motion. 1

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S. TINDEL AND J. UNTERBERGER

ANALYTIC FRACTIONAL BROWNIAN MOTION

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