×

On the construction of Wiener integrals with respect to certain pseudo-Bessel processes. (English) Zbl 1110.60055

Let \((B_t)\) be a standard real Brownian motion, \((R_t)\) be a \((B_t)\)-driven Bessel process, and \(B_t^{(s)}:=\frac{1}{\sqrt s} B_{st}\). In order to define stochastic integrals such as \(\int^1_0h(s)d [\sqrt sf(\frac{Bs}{\sqrt s})]\) or \(\int^1_0h(s)d[\sqrt sf(\frac{Rs}{\sqrt s})]\), for \(h\in L^2[0,1]\), the authors are led to consider the existence of integrals \(\int^1_0h(s)_xF[B^{(s)}]\frac {ds}{\sqrt s}\), where \(F\) is some Brownian functional. They find that this existence is in turn guaranteed by the finiteness of \(C^*_F:=\int^1_0|\mathbb{E} [F(B)_xF (B^{(s)})]|\frac{ds}{s}\). They study some sufficient conditions ensuring \(C^*_F<\infty\), and apply them in particular to the examples: \(F(B)= g(B_1)\) and \(F(B)=g(R_1)\).

MSC:

60H05 Stochastic integrals
60J65 Brownian motion
60J60 Diffusion processes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses, 10, Soc. Math. France, 2000; C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques, Panoramas et Synthèses, 10, Soc. Math. France, 2000 · Zbl 0982.46026
[2] Carmona, Ph.; Petit, F.; Yor, M., Beta-gamma random variables and interwining relations between certain Markov processes, Rev. Mat. Iberoamericana, 14, 2, 311-367 (1998) · Zbl 0919.60074
[3] Chen, L. H.Y., Poincaré-type inequalities via stochastic integrals, Zeit. für Wahr., 69, 251-277 (1985) · Zbl 0549.60019
[4] Ciesielski, Z.; Kerkyacharian, G.; Roynette, B., Quelques espaces fonctionnels associés à des processus gaussiens, Studia Math., 107, 2, 171-204 (1993) · Zbl 0809.60004
[5] Dym, H.; McKean, H. P., Gaussian Processes, Function Theory, and the Inverse Spectral Problem (1976), Academic Press · Zbl 0327.60029
[6] T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Process. Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Process. Related Fields (2006) (in press) · Zbl 1112.60042
[7] T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press) · Zbl 1112.60042
[8] T. Funaki, Y. Hariya, F. Hirsch, M. Yor, On some Fourier aspects of the construction of certain Wiener integrals (2005) (preprint); T. Funaki, Y. Hariya, F. Hirsch, M. Yor, On some Fourier aspects of the construction of certain Wiener integrals (2005) (preprint) · Zbl 1113.60055
[9] Gebelein, H., Das statistische problem der Korrelation als Variations und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung, Zeit. für Ang. Math. Mech., 21, 6, 364-379 (1941)
[10] Jeulin, T.; Yor, M., Inégalité de Hardy, semi-martingales et faux-amis, (Sém. Prob. XIII. Sém. Prob. XIII, Lect. Notes in Maths., vol. 721 (1979), Springer), 332-359 · Zbl 0419.60049
[11] Lamperti, J., Semi-stable Markov processes, I, Zeit. für Wahr., 22, 205-225 (1972) · Zbl 0274.60052
[12] Lebedev, N. N., Special Functions and their Applications (1972), Dover Publications · Zbl 0271.33001
[13] Ledoux, M., Inégalités isopérimétriques et calcul stochastique, (Sém. Prob. XXII. Sém. Prob. XXII, Lect. Notes in Maths, vol. 1321 (1988), Springer), 249-259 · Zbl 0649.60003
[14] Letac, G., Intégration et Probabilités, Analyse de Fourier. Exercices corrigés (1997), Masson · Zbl 0878.00004
[15] Mazet, O., Classification des semi-groupes de diffusion sur \(R\) associés à une famille de polynômes orthogonaux, (Sém. Prob. XXXI. Sém. Prob. XXXI, Lect. Notes in Maths, vol. 1655 (1997), Springer), 40-53 · Zbl 0883.60072
[16] Revuz, D.; Yor, M., Continuous Martingales and Brownian Motion (2005), Springer · Zbl 1087.60040
[17] Schoutens, W., (Stochastic Processes and Orthogonal Polynomials. Stochastic Processes and Orthogonal Polynomials, Lect. Notes in Stats., vol. 146 (2000), Springer) · Zbl 0960.60076
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.