Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series
stochastic convolutions; continuity of Gaussian processes; Gaussian trigonometric series
Summary:
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.
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. DOI 10.1111/1467-9965.00031 | MR 1446647
[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. MR 0510870
[3] M. Errami and F. Russo: Covariation de convolution de martingales. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 601–606. DOI 10.1016/S0764-4442(98)85014-3 | MR 1649341
[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. DOI 10.2307/2951677
[6] J.-P. Kahane: Some Random Series of Functions. 2nd ed., Cambridge University Press, Cambridge, 1985. MR 0833073 | Zbl 0571.60002
[7] J.-P. Kahane: Baire’s category theorem and trigonometric series. J. Anal. Math. 80 (2000), 143–182. DOI 10.1007/BF02791536 | MR 1771526 | 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. DOI 10.1007/BF02761823 | MR 0571394 | Zbl 0428.46035
Search
Partner of
