Séminaire de Probabilités XLVI pp 293-315

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

On Maximal Inequalities for Purely Discontinuous Martingales in Infinite Dimensions

Chapter

Abstract

The purpose of this paper is to give a survey of a class of maximal inequalities for purely discontinuous martingales, as well as for stochastic integral and convolutions with respect to Poisson measures, in infinite dimensional spaces. Such maximal inequalities are important in the study of stochastic partial differential equations with noise of jump type.

References

  1. 1.
    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge University Press, Cambridge, 2009). MR 2512800 (2010m:60002)Google Scholar
  2. 2.
    K. Bichteler, J.-B. Gravereaux, J. Jacod, Malliavin Calculus for Processes with Jumps (Gordon and Breach Science Publishers, New York, 1987). MR MR1008471 (90h:60056)Google Scholar
  3. 3.
    K. Bichteler, J. Jacod, Calcul de Malliavin pour les diffusions avec sauts: existence d’une densité dans le cas unidimensionnel, in Seminar on Probability, XVII. Lecture Notes in Math., vol. 986 (Springer, Berlin, 1983), pp. 132–157. MR 770406 (86f:60070)Google Scholar
  4. 4.
    Z. Brzeźniak, E. Hausenblas, J. Zhu, Maximal inequality of stochastic convolution driven by compensated Poisson random measures in Banach spaces, arXiv:1005.1600 (2010)Google Scholar
  5. 5.
    D.L. Burkholder, Distribution function inequalities for martingales. Ann. Probab. 1, 19–42 (1973). MR 0365692 (51 #1944)Google Scholar
  6. 6.
    D.L. Burkholder, R.F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124, 249–304 (1970). MR 0440695 (55 #13567)Google Scholar
  7. 7.
    V.H. de la Peña, E. Giné, Decoupling (Springer, New York, 1999). MR 1666908 (99k:60044)Google Scholar
  8. 8.
    S. Dirksen, Itô isomorphisms for L p-valued Poisson stochastic integrals. Ann. Probab. 42(6), 2595–2643 (2014). doi:10.1214/13-AOP906CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    S. Dirksen, J. Maas, and J. van Neerven, Poisson stochastic integration in Banach spaces, Electron. J. Probab. 18(100), 28 pp. (2013)Google Scholar
  10. 10.
    K. Dzhaparidze, E. Valkeila, On the Hellinger type distances for filtered experiments. Probab. Theory Relat. Fields 85(1), 105–117 (1990). MR 1044303 (91d:60102)Google Scholar
  11. 11.
    G. Fendler, Dilations of one parameter semigroups of positive contractions on L p spaces. Can. J. Math. 49(4), 736–748 (1997). MR MR1471054 (98i:47035)Google Scholar
  12. 12.
    A.M. Fröhlich, L. Weis, H calculus and dilations. Bull. Soc. Math. France 134(4), 487–508 (2006). MR 2364942 (2009a:47091)Google Scholar
  13. 13.
    G.H. Hardy, J.E. Littlewood, G.Pólya, Inequalities, 2nd edn. (Cambridge University Press, Cambridge, 1988). MR 0046395 (13,727e)Google Scholar
  14. 14.
    E. Hausenblas, J. Seidler, A note on maximal inequality for stochastic convolutions. Czech. Math. J. 51(126)(4), 785–790 (2001). MR MR1864042 (2002j:60092)Google Scholar
  15. 15.
    E. Hausenblas, J. Seidler, Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability. Stoch. Anal. Appl. 26(1), 98–119 (2008). MR 2378512 (2009a:60066)Google Scholar
  16. 16.
    J. Jacod, Th.G. Kurtz, S. Méléard, Ph. Protter, The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Stat. 41(3), 523–558 (2005). MR MR2139032 (2005m:60149)Google Scholar
  17. 17.
    S.G. Kreĭn, Yu.Ī. Petunı̄n, E.M. Semënov, Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54 (American Mathematical Society, Providence, 1982). MR 649411 (84j:46103)Google Scholar
  18. 18.
    H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in Real and Stochastic Analysis (Birkhäuser Boston, Boston, 2004), pp. 305–373. MR 2090755 (2005h:60169)Google Scholar
  19. 19.
    S. Kwapień, W.A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple (Birkhäuser, Boston, 1992). MR 1167198 (94k:60074)Google Scholar
  20. 20.
    M. Ledoux, M. Talagrand, Probability in Banach Spaces (Springer, Berlin, 1991). MR 1102015 (93c:60001)Google Scholar
  21. 21.
    E. Lenglart, Relation de domination entre deux processus. Ann. Inst. H. Poincaré Sect. B (N.S.) 13(2), 171–179 (1977). MR 0471069 (57 #10810)Google Scholar
  22. 22.
    E. Lenglart, D. Lépingle, M. Pratelli, Présentation unifiée de certaines inégalités de la théorie des martingales, in Séminaire de Probabilités, XIV (Paris, 1978/1979). Lecture Notes in Math., vol. 784 (Springer, Berlin, 1980), pp. 26–52. MR 580107 (82d:60087)Google Scholar
  23. 23.
    C. Marinelli, On maximal inequalities for purely discontinuous L q -valued martingales. Arxiv:1311.7120v1 (2013)Google Scholar
  24. 24.
    C. Marinelli, On regular dependence on parameters of stochastic evolution equations, in preparationGoogle Scholar
  25. 25.
    C. Marinelli, Local well-posedness of Musiela’s SPDE with Lévy noise. Math. Finance 20(3), 341–363 (2010). MR 2667893Google Scholar
  26. 26.
    C. Marinelli, Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps. J. Funct. Anal. 264(12), 2784–2816 (2013). MR 3045642Google Scholar
  27. 27.
    C. Marinelli, C. Prévôt, M.Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 258(2), 616–649 (2010). MR MR2557949Google Scholar
  28. 28.
    C. Marinelli, M. Röckner, On uniqueness of mild solutions for dissipative stochastic evolution equations. Infinite Dimens. Anal. Quantum Probab. Relat. Top. 13(3), 363–376 (2010). MR 2729590 (2011k:60220)Google Scholar
  29. 29.
    C. Marinelli, M. Röckner, Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise. Electron. J. Probab. 15(49), 1528–1555 (2010). MR 2727320Google Scholar
  30. 30.
    C. Marinelli, M. Röckner, On the maximal inequalities of Burkholder, Davis and Gundy, arXiv preprint (2013)Google Scholar
  31. 31.
    M. Métivier, Semimartingales (Walter de Gruyter & Co., Berlin, 1982). MR MR688144 (84i:60002)Google Scholar
  32. 32.
    P.A. Meyer, Le dual de H 1 est BMO (cas continu), Séminaire de Probabilités, VII (Univ. Strasbourg). Lecture Notes in Math., vol. 321 (Springer, Berlin, 1973), pp. 136–145. MR 0410910 (53 #14652a)Google Scholar
  33. 33.
    A.A. Novikov, Discontinuous martingales. Teor. Verojatnost. i Primemen. 20, 13–28 (1975). MR 0394861 (52 #15660)Google Scholar
  34. 34.
    Sz. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise (Cambridge University Press, Cambridge, 2007). MR MR2356959Google Scholar
  35. 35.
    Io. Pinelis, Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22(4), 1679–1706 (1994). MR 1331198 (96b:60010)Google Scholar
  36. 36.
    C. Prévôt (Knoche), Mild solutions of SPDE’s driven by Poisson noise in infinite dimensions and their dependence on initial conditions, Ph.D. thesis, Universität Bielefeld, 2005Google Scholar
  37. 37.
    Ph. Protter, D. Talay, The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25(1), 393–423 (1997). MR MR1428514 (98c:60063)Google Scholar
  38. 38.
    H.P. Rosenthal, On the subspaces of L p (p > 2) spanned by sequences of independent random variables. Isr. J. Math. 8, 273–303 (1970). MR 0271721 (42 #6602)Google Scholar
  39. 39.
    E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory (Princeton University Press, Princeton, 1970). MR 0252961 (40 #6176)Google Scholar
  40. 40.
    B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic analysis of operators on Hilbert space, 2nd edn. (Springer, New York, 2010). MR 2760647 (2012b:47001)Google Scholar
  41. 41.
    M. Veraar, L. Weis, A note on maximal estimates for stochastic convolutions. Czech. Math. J. 61(136)(3), 743–758 (2011). MR 2853088Google Scholar
  42. 42.
    L. Weis, The H holomorphic functional calculus for sectorial operators—a survey, in Partial Differential Equations and Functional Analysis (Birkhäuser, Basel, 2006), pp. 263–294. MR 2240065 (2007c:47018)Google Scholar
  43. 43.
    A.T.A. Wood, Rosenthal’s inequality for point process martingales. Stoch. Process. Appl. 81(2), 231–246 (1999). MR 1694561 (2000f:60073)Google Scholar
  44. 44.
    A.T.A. Wood, Acknowledgement of priority. Comment on: Rosenthal’s inequality for point process martingales. Stoch. Process. Appl. 81(2), 231–246 (1999). MR1694561 (2000f:60073); Stoch. Process. Appl. 93(2), 349 (2001). MR 1828780Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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