Electronic Journal of Probability

Poisson stochastic integration in Banach spaces

Sjoerd Dirksen, Jan Maas, and Jan Neerven

Full-text: Open access

Abstract

We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark- Ocone representation formula.

Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 100, 28 pp.

Dates
Accepted: 18 November 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064325

Digital Object Identifier
doi:10.1214/EJP.v18-2945

Mathematical Reviews number (MathSciNet)
MR3141801

Zentralblatt MATH identifier
1285.60049

Subjects
Primary: 60H05: Stochastic integrals
Secondary: 60G55: Point processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Stochastic integration Poisson random measure martingale type UMD Banach spaces stochastic convolutions Malliavin calculus Clark-Ocone representation theorem

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Dirksen, Sjoerd; Maas, Jan; Neerven, Jan. Poisson stochastic integration in Banach spaces. Electron. J. Probab. 18 (2013), paper no. 100, 28 pp. doi:10.1214/EJP.v18-2945. https://projecteuclid.org/euclid.ejp/1465064325


Export citation

References

  • Albiac, Fernando; Kalton, Nigel J. Topics in Banach space theory. Graduate Texts in Mathematics, 233. Springer, New York, 2006. xii+373 pp. ISBN: 978-0387-28141-4; 0-387-28141-X
  • Aronszajn, N.; Gagliardo, E. Interpolation spaces and interpolation methods. Ann. Mat. Pura Appl. (4) 68 1965 51–117.
  • Bourgain, Jean. Vector-valued singular integrals and the $H^ 1$-BMO duality. Probability theory and harmonic analysis (Cleveland, Ohio, 1983), 1–19, Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York, 1986.
  • Brzeźniak, Zdzisław; Hausenblas, Erika. Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Related Fields 145 (2009), no. 3-4, 615–637.
  • Burkholder, D. L. Distribution function inequalities for martingales. Ann. Probability 1 (1973), 19–42.
  • Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), no. 6, 997–1011.
  • 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.
  • Carlen, Eric A.; Pardoux, Étienne. Differential calculus and integration by parts on Poisson space. Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988), 63–73, Math. Appl., 59, Kluwer Acad. Publ., Dordrecht, 1990.
  • Cinlar, Erhan. Probability and stochastics. Graduate Texts in Mathematics, 261. Springer, New York, 2011. xiv+557 pp. ISBN: 978-0-387-87858-4
  • Clément, P.; de Pagter, B.; Sukochev, F. A.; Witvliet, H. Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no. 2, 135–163.
  • Dermoune, A.; Krée, P.; Wu, L. Calcul stochastique non adapté par rapport à la mesure aléatoire de Poisson. (French) [Nonadapted stochastic calculus with respect to the Poisson random measure] Séminaire de Probabilités, XXII, 477–484, Lecture Notes in Math., 1321, Springer, Berlin, 1988.
  • G. Di Nunno, B. Oksendal, and F. Proske, Malliavin calculus for Lévy processes with applications to finance, Universitext, Springer-Verlag, Berlin, 2009. (2010f:60001)
  • S. Dirksen, Itô isomorphisms for L^p-valued Poisson stochastic integrals, Arxiv 1208.3885.
  • Fendler, Gero. Dilations of one parameter semigroups of positive contractions on $L^ p$ spaces. Canad. J. Math. 49 (1997), no. 4, 736–748.
  • Frühlich, Andreas M.; Weis, Lutz. $H^ \infty$ calculus and dilations. Bull. Soc. Math. France 134 (2006), no. 4, 487–508.
  • Hausenblas, Erika. Maximal inequalities of the Itô integral with respect to Poisson random measures or Lévy processes on Banach spaces. Potential Anal. 35 (2011), no. 3, 223–251.
  • Hausenblas, Erika; Seidler, Jan. A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51(126) (2001), no. 4, 785–790.
  • Junge, Marius; Xu, Quanhua. Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31 (2003), no. 2, 948–995.
  • Kabanov, Ju. M. Extended stochastic integrals. (Russian) Teor. Verojatnost. i Primenen. 20 (1975), no. 4, 725–737.
  • Kunita, Hiroshi. Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. Real and stochastic analysis, 305–373, Trends Math., Birkhäuser Boston, Boston, MA, 2004.
  • Last, Günter; Penrose, Mathew D. Martingale representation for Poisson processes with applications to minimal variance hedging. Stochastic Process. Appl. 121 (2011), no. 7, 1588–1606.
  • Last, Günter; Penrose, Mathew D. Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Related Fields 150 (2011), no. 3-4, 663–690.
  • Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
  • Løkka, Arne. Martingale representation of functionals of Lévy processes. Stochastic Anal. Appl. 22 (2004), no. 4, 867–892.
  • Maas, Jan. Malliavin calculus and decoupling inequalities in Banach spaces. J. Math. Anal. Appl. 363 (2010), no. 2, 383–398.
  • Maas, Jan; van Neerven, Jan. A Clark-Ocone formula in UMD Banach spaces. Electron. Commun. Probab. 13 (2008), 151–164.
  • Marinelli, Carlo; Prévôt, Claudia; Röckner, Michael. Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 258 (2010), no. 2, 616–649.
  • Marinelli, Carlo; Röckner, Michael. Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise. Electron. J. Probab. 15 (2010), no. 49, 1528–1555.
  • Maurey, B. Système de Haar. (French) Séminaire Maurey-Schwartz 1974-1975: Espaces L$\sup{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. I et II, 26 pp. (erratum, p. 1) Centre Math., École Polytech., Paris, 1975.
  • J.M.A.M. van Neerven, Stochastic evolutions equations, 2007, Lecture notes of the 2007 Internet Seminar, available at repository.tudelft.nl.
  • 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.
  • 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
  • Nualart, David; Vives, Josep. Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités, XXIV, 1988/89, 154–165, Lecture Notes in Math., 1426, Springer, Berlin, 1990.
  • Pisier, Gilles. Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), no. 3-4, 326–350.
  • Privault, Nicolas. Chaotic and variational calculus in discrete and continuous time for the Poisson process. Stochastics Stochastics Rep. 51 (1994), no. 1-2, 83–109.
  • Rosiński, J. Random integrals of Banach space valued functions. Studia Math. 78 (1984), no. 1, 15–38.
  • J.L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Probability and Banach spaces (Zaragoza, 1985), Lecture Notes in Math., vol. 1221, Springer, Berlin, 1986, pp. 195–222. (88g:42020)
  • Rüdiger, B. Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces. Stoch. Stoch. Rep. 76 (2004), no. 3, 213–242.
  • Stein, Elias M. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63 Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1970 viii+146 pp.
  • M. Veraar, Stochastic integration in Banach spaces and applications to parabolic evolution equations, Ph.D. thesis, Delft University of Technology, 2006.
  • Wu, Liming. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000), no. 3, 427–438.
  • J. Zhu. Maximal inequalities for stochastic convolutions driven by Lévy processes in Banach spaces, Work in progress.