Continuity of stochastic convolutions. (English) Zbl 1001.60056

Summary: Let \(B\) be a Brownian motion, and let \(\mathcal C_{p}\) be the space of all continuous periodic functions \(f\:\mathbb R\to \mathbb R\) with period 1. It is shown that the set of all \(f\in \mathcal C_{p}\) such that the stochastic convolution \(X_{f,B}(t)= \int _0^t f(t-s) d B(s)\), \(t\in [0,1]\), does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.


60H05 Stochastic integrals
60G15 Gaussian processes
60G17 Sample path properties
60G50 Sums of independent random variables; random walks
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