ROUGH VOLTERRA EQUATIONS THE ALGEBRAIC

ROUGH VOLTERRA EQUATIONS 1: THE ALGEBRAIC INTEGRATION SETTING. AURÉLIEN DEYA AND SAMY TINDEL Abstract. We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with Hölder exponent > 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coe?cient. In case of a driving signal with Hölder exponent 1/3 < ≤ 1/2, we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coe?cient H > 1/3. 1. Introduction This article is the first of a series of two papers dealing with Volterra equations driven by rough paths. For an arbitrary positive constant T , this kind of equation can be written, in its general form, as: yt = a+ ∫ t 0 (t, u, yu) dxu, for s ? [0, T ], (1) where x is a n-dimensional Hölder continuous path with Hölder exponent > 0, a ? ?d stands for an initial condition, and : ?+??+??d ? ?d,n is a smooth enough function. Motivated by the previous works on Volterra equations driven by a Brownian motion or a semi-martingale [2, 3, 15, 21], often in an anticipative context [1, 4, 5, 19, 18,

• ?su ?


• unique ?? ?


• then


• dimensional hölder


• algebraic integration


• rough volterra


• hölder exponent


• smooth functions


• called algebraic