Rough Volterra equations 2: Convolutional generalized integrals

We define and solve Volterra equations driven by a non-differentiable signal, by means of a variant of the rough paths theory which allows us to handle generalized integrals weighted by an exponential coefficient. The results are applied to a standard rough path x = ( x 1 , x 2 ) ∈ C 2 γ ( R m ) × C 2 2 γ ( R m , m ) , with γ > 1 / 3 , which includes the case of fractional Brownian motion with Hurst index H > 1 / 3 .