ROUGH VOLTERRA EQUATIONS CONVOLUTIONAL GENERALIZED INTEGRALS

ROUGH VOLTERRA EQUATIONS 2: CONVOLUTIONAL GENERALIZED INTEGRALS AURÉLIEN DEYA AND SAMY TINDEL Abstract. We define and solve Volterra equations driven by a non-differentiable signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to a standard rough path x = (x1,x2) ? C?2 (R m)?C2?2 (R m,m), with ? > 1/3, which includes the case of fractional Brownian motion with Hurst index H > 1/3. 1. Introduction This paper is part of an ambitious ongoing project which aims at offering a new point of view on multidimensional stochastic calculus, via the semi-deterministic rough path approach initiated by Lyons [24]. We tackle the issue of the non-linear Volterra system yit = a i + ∫ t 0 ?i0(t, u, yu) du+ m∑ j=1 ∫ t 0 ?ij(t, u, yu) dx j u, i = 1, . . . , d, t ? [0, T ], (1) where T stands for an arbitrary horizon, x : [0, T ] ? Rm a multidimensional ?-Hölder path, a ? Rd an initial condition and ?ij : [0, T ]2 ? Rd ? R smooth enough functions.

• global solution


• volterra equations driven


• rough paths


• paths theory


• volterra system perturbed


• system


• partial differential equations


• continuous functions


• volterra equation


• hts ?