Continuity of stochastic convolutions

Zdzisław Brzeźniak; Szymon Peszat; Jerzy Zabczyk

Czechoslovak Mathematical Journal (2001)

  • Volume: 51, Issue: 4, page 679-684
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Brzeźniak, Zdzisław, Peszat, Szymon, and Zabczyk, Jerzy. "Continuity of stochastic convolutions." Czechoslovak Mathematical Journal 51.4 (2001): 679-684. <http://eudml.org/doc/30664>.

@article{Brzeźniak2001,
abstract = {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.},
author = {Brzeźniak, Zdzisław, Peszat, Szymon, Zabczyk, Jerzy},
journal = {Czechoslovak Mathematical Journal},
keywords = {stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series; stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series},
language = {eng},
number = {4},
pages = {679-684},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuity of stochastic convolutions},
url = {http://eudml.org/doc/30664},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Brzeźniak, Zdzisław
AU - Peszat, Szymon
AU - Zabczyk, Jerzy
TI - Continuity of stochastic convolutions
JO - Czechoslovak Mathematical Journal
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 4
SP - 679
EP - 684
AB - 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.
LA - eng
KW - stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series; stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series
UR - http://eudml.org/doc/30664
ER -

References

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