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

top
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

top

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

top
  1. 10.1111/1467-9965.00031, Math. Finance 7 (1997), 211–239. (1997) MR1446647DOI10.1111/1467-9965.00031
  2. Sur les coefficients de Fourier des fonctions continues, C. R.  Acad. Sci. Paris Sér. A-B 285 (1977), A1001–A1003. (1977) MR0510870
  3. 10.1016/S0764-4442(98)85014-3, C.  R.  Acad. Sci. Paris Sér. I Math. 326 (1998), 601–606. (1998) MR1649341DOI10.1016/S0764-4442(98)85014-3
  4. On Stochastic Convolutions, Report S98–19, School of Mathematics, University of New South Wales, Sydney, 1998. (1998) 
  5. 10.2307/2951677, Econometrica 60 (1992), 77–105. (1992) DOI10.2307/2951677
  6. Some Random Series of Functions, 2nd ed., Cambridge University Press, Cambridge, 1985. (1985) Zbl0571.60002MR0833073
  7. 10.1007/BF02791536, J.  Anal. Math. 80 (2000), 143–182. (2000) Zbl0961.42001MR1771526DOI10.1007/BF02791536
  8. Stochastic PDEs and term structure models, Journees Internationales des Finance, IGR-AFFI, La Boule, 1993. (1993) 
  9. 10.1007/BF02761823, Israel J.  Math. 34 (1979), 38–44. (1979) Zbl0428.46035MR0571394DOI10.1007/BF02761823

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.