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.

MSC:

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

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