Abstract
In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<frac{1}{2}. Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.
Highlights
Consider the d-dimensional stochastic differential equation (SDE) x+ α Lt (X x) t = x+ α Lt (X x ) · 1d + BtH,0 ≤ t ≤ T, x ∈ Rd, (1.1)where the driving noise B·H of this equation is a d-dimensional fractional Brownian motion, whose components are given by one-dimensional independent fractionalBrownian motions with a Hurst parameter H ∈ (0, 1/2), and where α ∈ R is a constant and 1d is the vector in Rd with entries given by 1
Where the driving noise B·H of this equation is a d-dimensional fractional Brownian motion, whose components are given by one-dimensional independent fractional
Our paper is organized as follows: In Sect. 2, we introduce the framework of our paper and recall in this context some basic facts from fractional calculus and Malliavin calculus for Brownian noise
Summary
Where the driving noise B·H of this equation is a d-dimensional fractional Brownian motion, whose components are given by one-dimensional independent fractional. The method of the authors for the construction of unique weak solutions of such equations is based on the construction of a certain resolvent family on the space Cb(Rd ) in connection with the properties of the Kato class Kd−1 In this context, we mention the paper [20] on SDE’s with distributional drift coefficients. We want to point out a recent work of Catellier, Gubinelli [11], which came to our attention, after having finalized our article In their striking paper, which extends the results of Davie [15] to the case of a fractional Brownian noise, the authors study the problem, which fractional Brownian paths regularize solutions to SDE’s of the form d.
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