×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] T. Bjork, Y. Kabanov and W. Runggaldier: Bond market structure in the presence of marked point processes. Math. Finance 7 (1997), 211-239. · Zbl 0884.90014
[2] K. De Leeuw, J.-P. Kahane and Y. Katznelson: Sur les coefficients de Fourier des fonctions continues. C. R. Acad. Sci. Paris Sér. A-B 285 (1977), A1001-A1003. · Zbl 0372.42004
[3] M. Errami and F. Russo: Covariation de convolution de martingales. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 601-606. · Zbl 0917.60054
[4] B. Goldys and M. Musiela: On Stochastic Convolutions. Report S98-19, School of Mathematics, University of New South Wales, Sydney, 1998.
[5] D. Heath, A. Jarrow and A. Morton: Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation. Econometrica 60 (1992), 77-105. · Zbl 0751.90009
[6] J.-P. Kahane: Some Random Series of Functions. 2nd, Cambridge University Press, Cambridge, 1985. · Zbl 0571.60002
[7] J.-P. Kahane: Baire’s category theorem and trigonometric series. J. Anal. Math. 80 (2000), 143-182. · Zbl 0961.42001
[8] M. Musiela: Stochastic PDEs and term structure models. Journees Internationales des Finance, IGR-AFFI, La Boule, 1993.
[9] G. Pisier: A remarkable homogeneous Banach algebra. Israel J. Math. 34 (1979), 38-44. · Zbl 0428.46035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.