Poisson stochastic integration in Banach spaces

Sjoerd Dirksen (University of Bonn)
Jan Maas (University of Bonn)
Jan van Neerven (Delft University of Technology)

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.


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Pages: 1-28

Publication Date: November 18, 2013

DOI: 10.1214/EJP.v18-2945

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 MR2192298
  • Aronszajn, N.; Gagliardo, E. Interpolation spaces and interpolation methods. Ann. Mat. Pura Appl. (4) 68 1965 51--117. MR0226361
  • 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. MR0830227
  • Brzeźniak, Zdzisław; Hausenblas, Erika. Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Related Fields 145 (2009), no. 3-4, 615--637. MR2529441
  • Burkholder, D. L. Distribution function inequalities for martingales. Ann. Probability 1 (1973), 19--42. MR0365692
  • Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), no. 6, 997--1011. MR0632972
  • 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
  • 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. MR1052702
  • Çınlar, Erhan. Probability and stochastics. Graduate Texts in Mathematics, 261. Springer, New York, 2011. xiv+557 pp. ISBN: 978-0-387-87858-4 MR2767184
  • Clément, P.; de Pagter, B.; Sukochev, F. A.; Witvliet, H. Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), no. 2, 135--163. MR1749077
  • 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. MR0960543
  • G. Di Nunno, B. Oksendal, and F. Proske, phMalliavin calculus for Lévy processes with applications to finance, Universitext, Springer-Verlag, Berlin, 2009. MR2460554 (2010f:60001)
  • S. Dirksen, phItô 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. MR1471054
  • Fröhlich, Andreas M.; Weis, Lutz. $H^ \infty$ calculus and dilations. Bull. Soc. Math. France 134 (2006), no. 4, 487--508. MR2364942
  • 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. MR2832576
  • Hausenblas, Erika; Seidler, Jan. A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51(126) (2001), no. 4, 785--790. MR1864042
  • Junge, Marius; Xu, Quanhua. Noncommutative Burkholder/Rosenthal inequalities. Ann. Probab. 31 (2003), no. 2, 948--995. MR1964955
  • Kabanov, Ju. M. Extended stochastic integrals. (Russian) Teor. Verojatnost. i Primenen. 20 (1975), no. 4, 725--737. MR0397877
  • 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. MR2090755
  • 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. MR2802467
  • 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. MR2824870
  • 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 MR1102015
  • Løkka, Arne. Martingale representation of functionals of Lévy processes. Stochastic Anal. Appl. 22 (2004), no. 4, 867--892. MR2062949
  • Maas, Jan. Malliavin calculus and decoupling inequalities in Banach spaces. J. Math. Anal. Appl. 363 (2010), no. 2, 383--398. MR2564861
  • Maas, Jan; van Neerven, Jan. A Clark-Ocone formula in UMD Banach spaces. Electron. Commun. Probab. 13 (2008), 151--164. MR2399277
  • 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. MR2557949
  • 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. MR2727320
  • 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. MR0420839
  • J.M.A.M. van Neerven, phStochastic 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. MR2330977
  • 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
  • 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. MR1071538
  • Pisier, Gilles. Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), no. 3-4, 326--350. MR0394135
  • 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. MR1380764
  • Rosiński, J. Random integrals of Banach space valued functions. Studia Math. 78 (1984), no. 1, 15--38. MR0766703
  • J.L. Rubio de Francia, phMartingale 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. MRMR875011 (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. MR2072381
  • 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. MR0252961
  • 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. MR1800540
  • J. Zhu, Maximal inequalities for stochastic convolutions driven by Lévy processes in Banach spaces, Work in progress.
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