Stochastic convolutions driven by martingales: Maximal inequalities and exponential integrability

Stochastic Analysis and Applications

Volumn 26, Issue 1, 2008, Pages 98-119

  Hausenblas, Erika ^a   Seidler, Jan ^b  

^a UNIVERSITY OF SALZBURG   (Austria)

^b INSTITUTE OF MATHEMATICS   (Czech Republic)

Author keywords

Exponential tail estimates; Maximal inequality; Stochastic convolutions; Stochastic partial differential equations; Unitary dilations

Indexed keywords

EID: 37749031469     PISSN: 07362994     EISSN: 15329356     Source Type: Journal    
DOI: 10.1080/07362990701673047     Document Type: Article

References (42)

1. Albeverio, S., Wu, J.-L., and Zhang, T.-S. 1998. Parabolic SPDEs driven by Poisson white noise. Stochastic Processes and Applications 74:21-36.
2. Applebaum, D., and Wu, J.-L. 2000. Stochastic partial differential equations driven by Levy space-time white noise. Random Operations and Stochastic Equations 8:245-259.
3. Asperti, C. 1989. Sur l'intégration stochastique par rapport à une martingale hilbertienne de carré intégrable. Probability and Theory of Related Fields 81:139-158.
4. Badrikian, A. 1996. Martingales hilbertiennes. Annals of Mathematics Blaise Pascal Numéro Spécial: 115-171.
5. Blount, D., and Kouritzin, M.A. 2001. Holder continuity for spatial and path processes via spectral analysis. Probability and Theory Related of Fields 119:589-603.
6. Brzeźniak, Z. 1997. On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61:245-295.
7. Brzeźniak, Z., and Peszat, S. 2000. Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces. In Stochastic Processes, Physics and Geometry: New Interplays (Leipzig, 1999), Vol. I. Amer. Math. Soc., Providence, pp. 55-64.
8. Burkholder, D.L. 1988. Sharp inequalities for martingales and stochastic integrals. Astérisque 158-159:75-94.
9. Da Prato, G., Kwapień, S., and Zabczyk, J. 1987. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23:1-23.
10. Da Prato, G., and Zabczyk, J. 1992. A note on stochastic convolution. Stochastic Analysis and Applications 10:143-153.
11. Da Prato, G., and Zabczyk, J. 1992. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge.
12. Davies, E.B. 1976. Quantum Theory of Open Systems. Academic Press, London.
13. p norms of stochastic integrals and other martingales. Duke Math. J. 43:697-704.
14. Dellacherie, C., and Meyer, P.-A. 1980. Probabilités et Potentiel. Chapitres V àVIII. Hermann, Paris.
15. Fendler, G. 1997. Dilations of one parameter semigroups of positive contractions on U spaces. Canadian Journal of Mathematics 49:736-748.
16. Fournier, N. 2000. Malliavin calculus for parabolic SPDEs with jumps. Stochastic Processes and Applications 87:115-147.
17. Fuhrman, M., and Röckner, M. 2000. Generalized Mehler semigroups: The non-Gaussian case. Potential Analysis 12:1-47.
18. Goldys, B. 1990. On some regularity properties of solutions to stochastic evolution equations in Hilbert spaces. Colloquium Mathematicum 58:327-338.
19. Gyöngy, I., and Krylov, N.V. 1980/81. On stochastic equations with respect to semimartingales I. Stochastics 4:1-21.
20. Hamedani, H.D., and Zangeneh, B.J. 2001. Stopped Doob inequality for pth moment, 0 < p < ∞, stochastic convolution integrals. Stochastic Analysis and Applications 19:771-798.
21. Hausenblas, E. 2005. Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure. Electronic Journal of Probability 10:1496-1546.
22. Hausenblas, E., and Seidler, J. 2001. A note on maximal inequality for stochastic convolutions. Czechoslovak Mathematical Journal 51:785-790.
23. Ichikawa, A. 1986. Some inequalities for martingales and stochastic convolutions. Stochastic Analysis and Applications 4:329-339.
24. Kallenberg, O., and Sztencel, R. 1991. Some dimension-free features of vector-valued martingales. Probabability Theory Related Fields 88:215-247.
25. Knoche, C. 2005. Mild Solutions of SPDE's Driven by Poisson Noise in Infinite Dimensions and Their Dependence on Initial Conditions. Dissertation, Universität Bielefeld, (BiBoS Preprint E05-10-194).
26. Kotelenez, P. 1982. A submartingale type inequality with applications to stochastic evolution equations. Stochastics 8:139-151.
27. Kotelenez, P. 1984. A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations. Stochastic Analysis and Applications 2:245-265.
28. Kotelenez, P. 1987. A maximal inequality for stochastic convolution integrals on Hilbert spaces and space-time regularity of linear stochastic partial differential equations. Stochastics 21:345-358.
29. Métivier, M. 1982. Semimartingales. Walter de Gruyter, Berlin.
30. Metivier, M., and Pellaumail, J. 1976. Cylindrical stochastic integral. Rapport No 63, I.R.I.S.A. Rennes.
31. Metivier, M., and Pellaumail, J. 1980. On a stopped Doob's inequality and general stochastic equations. Annals of Probability 8:96-114.
32. Metivier, M., and Pellaumail, J. 1980. Stochastic Integration. Academic Press, New York.
33. Ondreját, M. 2004. Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Mathematicae 426:1-63.
34. Ondreját, M. 2005. Brownian representations of cylindrical martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces. Czechoslovak Mathematical Journal 55:1003-1039.
35. Peszat, S., and Seidler, J. 1998. Maximal inequalities and space-time regularity of stochastic convolutions. Mathematica Bohemica 123:7-32.
36. Peszat, S., and Zabczyk, J. 2006. Stochastic heat and wave equations driven by an impulsive noise. In Stochastic Partial Differential Equations and Applications VII. (Trento, 2004). Chapman & Hall/CRC, Boca Raton, pp. 229-242.
37. Rogers, L.C.G., and Williams, D. 1994. Diffusions, Markov Processe, and Martingales. Vol. 2, Itô calculus. 2nd Ed. Cambridge University Press, Cambridge.
38. Seidler, J. 1993. Da Prato-Zabczyk's maximal inequality revisited I. Mathematica Bohémica 118:67-106.
39. Seidler, J., and Sobukawa, T. 2003. Exponential integrability of stochastic convolutions. Journal of London Mathematical Society (2) 67:245-258.
40. Shiryaev, A.N., and Chernyl, A.S. 2002. A vector stochastic integral and the fundamental theorem of asset pricing (Russian). Trudy Matematicheskogo Instituto im. V A. Steklova 237:12-56.
41. Sz.-Nagy, B., and Foiaş, C. 1970. Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam.
42. Tubaro, L. 1984. An estimate of Burkholder type for stochastic processes defined by the stochastic integral. Stochastic Analysis and Applications 2:187-192.