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Continuity of stochastic convolutions

Zdzisław Brzeźniak, Szymon Peszat, Jerzy Zabczyk (2001)

Czechoslovak Mathematical Journal

Let B be a Brownian motion, and let 𝒞 p be the space of all continuous periodic functions f ℝ → ℝ with period 1. It is shown that the set of all f ∈ 𝒞 p such that the stochastic convolution X f , B ( t ) = ∫ 0 t f ( t - s ) d B ( s ) , t ∈ [ 0 , 1 ] does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.

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