Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 13 (2008) > Paper 15 open journal systems 


A Clark-Ocone formula in UMD Banach spaces

Jan Maas, TU Delft
Jan van Neerven, TU Delft


Abstract
Let H be a separable real Hilbert space and let F = (Ft)t in [0,T] be the augmented filtration generated by an H-cylindrical Brownian motion WH on [0,T] on a probability space (Ω,F,P). We prove that if E is a UMD Banach space, 1≤ p<∞, and f in D1,p(E) is FT-measurable, then f = E f + int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of Lp-stochastically integrable L(H,E)-valued processes.


Full text: PDF

Pages: 151-164

Published on: April 7, 2008
Bibliography
  1. Aase, Knut; Øksendal, Bernt; Privault, Nicolas; Ubøe, Jan. White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance. Finance Stoch. 4 (2000), no. 4, 465--496. MR1779589 (2001j:60126)
  2. Bourgain, Jean. Vector-valued singular integrals and the $Hsp 1$-BMO duality. Probability theory and harmonic analysis (Cleveland, Ohio, 1983), 1--19, Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York, 1986. MR0830227 (87j:42049b)
  3. Burkholder, Donald L. Martingales and singular integrals in Banach spaces. Handbook of the geometry of Banach spaces, Vol. I, 233--269, North-Holland, Amsterdam, 2001. MR1863694 (2003b:46009)
  4. Carmona, René A.; Tehranchi, Michael R. Interest rate models: an infinite dimensional stochastic analysis perspective.Springer Finance. Springer-Verlag, Berlin, 2006. xiv+235 pp. ISBN: 978-3-540-27065-2; 3-540-27065-5 MR2235463 (2008a:91001)
  5. Clark, J. M. C. The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Statist. 41 1970 1282--1295. MR0270448 (42 #5336)
  6. Clément, P.; de Pagter, B.; Sukochev, F. A.; Witvliet, H. Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no. 2, 135--163. MR1749077 (2002c:47036)
  7. de Faria, Margarida; Oliveira, Maria João; Streit, Ludwig. A generalized Clark-Ocone formula. Random Oper. Stochastic Equations 8 (2000), no. 2, 163--174. MR1765875 (2001g:60159)
  8. Diestel, Joe; Jarchow, Hans; Tonge, Andrew. Absolutely summing operators.Cambridge Studies in Advanced Mathematics, 43. Cambridge University Press, Cambridge, 1995. xvi+474 pp. ISBN: 0-521-43168-9 MR1342297 (96i:46001)
  9. N.J. Kalton and L. Weis, The ${H}^infty$-functional calculus and square function estimates, in preparation.
  10. I. Karatzas, D.L. Ocone, and J. Li, An extension of Clark's formula, Stochastics Stochastics Rep. 37 (1991), no. 3, 127--131.
  11. Maas, J. Malliavin calculus and decoupling inequalities in Banach spaces, arXiv: 0801.2899v2 [math.FA], submitted for publication.
  12. Malliavin, Paul; Nualart, David. Quasi-sure analysis and Stratonovich anticipative stochastic differential equations. Probab. Theory Related Fields 96 (1993), no. 1, 45--55. MR1222364 (94c:60100)
  13. Mayer-Wolf, E.; Zakai, M. The Clark-Ocone formula for vector valued Wiener functionals. J. Funct. Anal. 229 (2005), no. 1, 143--154. MR2180077 (2006h:60092)
  14. Mayer-Wolf, E.; Zakai, M. The divergence of Banach space valued random variables on Wiener space. Probab. Theory Related Fields 132 (2005), no. 2, 291--320. MR2199294 (2007e:60039)
  15. van Neerven, J. M. A. M.; Veraar, M. C.; Weis, L. Stochastic integration in UMD Banach spaces. Ann. Probab. 35 (2007), no. 4, 1438--1478. MR2330977
  16. Nualart, David. The Malliavin calculus and related topics.Second edition.Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233 (2006j:60004)
  17. Ocone, Daniel. Malliavin's calculus and stochastic integral representations of functionals of diffusion processes. Stochastics 12 (1984), no. 3-4, 161--185. MR0749372 (85m:60101)
  18. Osswald, Horst. On the Clark Ocone formula for the abstract Wiener space. Adv. Math. 176 (2003), no. 1, 38--52. MR1978340 (2004a:60109)
  19. Pisier, Gilles. The volume of convex bodies and Banach space geometry.Cambridge Tracts in Mathematics, 94. Cambridge University Press, Cambridge, 1989. xvi+250 pp. ISBN: 0-521-36465-5; 0-521-66635-X MR1036275 (91d:52005)













Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X