Geodesic
Continuity of stochastic convolutions
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 679-684 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $B$ be a Brownian motion, and let $\mathcal C_{\mathrm p}$ be the space of all continuous periodic functions $f\:\mathbb{R}\rightarrow \mathbb{R}$ with period 1. It is shown that the set of all $f\in \mathcal C_{\mathrm p}$ such that the stochastic convolution $X_{f,B}(t)= \int _0^tf(t-s)\mathrm{d}B(s)$, $t\in [0,1]$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.
Let $B$ be a Brownian motion, and let $\mathcal C_{\mathrm p}$ be the space of all continuous periodic functions $f\:\mathbb{R}\rightarrow \mathbb{R}$ with period 1. It is shown that the set of all $f\in \mathcal C_{\mathrm p}$ such that the stochastic convolution $X_{f,B}(t)= \int _0^tf(t-s)\mathrm{d}B(s)$, $t\in [0,1]$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.
Classification : 60G15, 60G17, 60G50, 60H05
Keywords: stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series 
@article{CMJ_2001_51_4_a1,
     author = {Brze\'zniak, Zdzis{\l}aw and Peszat, Szymon and Zabczyk, Jerzy},
     title = {Continuity of stochastic convolutions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {679--684},
     year = {2001},
     volume = {51},
     number = {4},
     mrnumber = {1864035},
     zbl = {1001.60056},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a1/}
}
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Brzeźniak, Zdzisław; Peszat, Szymon; Zabczyk, Jerzy. Continuity of stochastic convolutions. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 4, pp. 679-684. http://geodesic.mathdoc.fr/item/CMJ_2001_51_4_a1/