Continuity of Stochastic Convolutions

Abstract

Let B be a Brownian motion, and let $\mathcal{C}_{\text{p}}$ be the space of all continuous periodic functions f ℝ→ℝ with period 1. It is shown that the set of all f→$\mathcal{C}_{\text{p}}$ such that the stochastic convolution $X_{f,B} (t) = \int_0^t {f(t - s){\text{d}}B(s),t \in [0,1]}$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.