Stochastic integrals and the Lévy-Ito decomposition theorem on separable Banach spaces

Stochastic Analysis and Applications

Volumn 23, Issue 2, 2005, Pages 217-253

  Albeverio, S ^a   Rudiger B ^b  

^a UNIVERSITY OF BONN   (Germany)

^b Universität Koblenz Landau ^*  (Germany)

Author keywords

C dl g processes on Banach spaces; Independent increments; L vy measures; L vy Ito decomposition; Random martingales measures; Stochastic integrals; Type 2 spaces

Indexed keywords

EID: 16244411110     PISSN: 07362994     EISSN: None     Source Type: Journal    
DOI: 10.1081/SAP-200026429     Document Type: Article

References (63)

1. Albeverio, S.; Höegh-Krohn, R.; Holden, H.; Kolsrud, T. White noise on vector bundels and local functionals, J. Funct. Anal. 1995, 134, 1-32.
2. Albeverio, S.; Rüdiger, B. 2002. Infinite dimensional stochastic differential equations obtained by subordination and related Dirichlet forms, Preprint BiBoS, 02-03-076, J. Funct. Anal. 2003, 204 (1), 122-156.
3. Araujo, A.; Giné, E. The Central Limit Theorem for Real and Banach Valued Random Variables; Wiley Series in Probability and Mathematical Statistics; New York, Chichester, Brisbane, Toronto, 1980.
4. Araujo, A.; Giné, E. Type, coptype and Lévy measures in Banach spaces. Ann. Probab. 1978, 6 (4), 637-643.
5. Applebaum, D.; Wu, J. L. Stochastic partial differential equations driven by Lévy space-time white noise. Random Oper. Stochast. Eqs. 2000, 8 (3), 245-259.
6. Albeverio, S.; Wu, J.-L.; Zhang, T.-S. Parabolic SPDEs driven by Poisson white noise. Stoch. Proc. Appl. 1998, 74, 21-36.
7. Badrikian, A. Prolégomènes au calcul des probabilités dans les Banach. In Ecole d'Été de Probabilités de Saint-Flour V-1975; Hennequin, P.L., Ed.; LNM 539; Springer: Berlin-Heidelberg-New York, 1976.
8. Bauer, H. Wahrscheinlichkeitstheorie und Grundzüge der Masstheorie; 2 edition de Gruyter Lehrbuch; Walter de Gruyter: Berlin-New York, 1974.
9. Barndorff-Nielsen, O.; Mikosch, T.; Resnik, S. Lëvy Processes: Theory and Applications; Birkhäuser Verlag: New York, 2001.
10. Bensoussan, A.; Lions, J.-L. Contrôle impulsionnel et inéquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities], pp. 445, in Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science]; Gauthier-Villars: Paris, 1982; Vol. 11.
11. Bertoin, J. Lévy Processes; Cambridge University Press: Cambridge, U.K., 1996.
12. Chung, K.L. A Course in Probability Theory, Probability and Mathematical Statistics; A series of Monographs and Textbooks; Academic Press: New York, San Francisco, London, 1974.
13. Dettweiler, E. Banach space valued processes with independent increments and stochastic integrals. In Probability in Banach spaces IV, Proc., Oberwolfach 1982, Lectures Notes Maths; Springer: Berlin, 1982; 54-83.
14. Dynkin, E.B. Die Grundlagen der Theorie der Markoffschen Prozesse; Springer Verlag: Berlin-Göttingen-Heidelberg-New York, 1982.
15. Fernique, X. Lois infiniment divisibles sur l'espace des distributions. Inv. Math. 1967, 3, 289-292.
16. Graveraux, J.B.; Pellaumail, J. Formule d'Ito pour des processes non continus à valeurs dans des espaces de Banach. Ann. Inst. H. Poincaré, Sect. B, 1974, 70, 399-422.
17. Hamedani, G.; Mandrekar, V. Lévy-Khinchine representation and Banach spaces of type and cotype. Stud. Math. 66 1979, 180 (3), 299-306.
18. Gihman, I.I.; Skorohod, A.V. The Theory of Stochastic Processes II; Springer: Berlin-Heidelberg-New York, 1975.
19. Hoffmann-Jörgensen, J. Probability in Banach spaces. In Ecole d' Eté de Probabilités de Saint-Flour VI-1976; Hennequin, P.L., Ed.; Lecture Notes in Math. 598; Springer Verlag: Berlin-Heidelberg-New York, 1977.
20. Ikeda, N.; Watanabe, S. Stochastic Differential Equations and Diffusion Processes, 2nd Ed.; North-Holland Mathematical Library; North-Holland Publishing Company: Amsterdam, Oxford, New York, 1989; Vol. 24.
21. Ito, K. On stochastic processes I (Infinitely divisible laws of probability). Jap. J. Math. 1942, 78, 261-301.
22. Ito, K. Continuous Additive script S sign-processes; Lecture Notes in Control and Inform. Sci.; Springer: New York, 1980; Vol. 25, 36-46.
23. Ito, K.; McKean, H.P. Diffusion Processes and Their Sample Paths; Die Grundlehren der Mathematischen Wissenschaften, Bd 125; Springer Verlag: Berlin-New York, 1974. Second printing, corrected.
24. Ito, K.; Nisio, M. On the convergence of sums of independent Banach space valued random variables. Osaka J. Math. 1968, 5, 35-48.
25. Jurek, Z.J.; Mason, J.D. Operator-Limit Distributions in Probability Theory; Wiley-Interscience Publication, John Wiley and Sons, Inc.: New York, Chichester, Brisbane, Toronto, Singapore, 1993.
26. Kallianpur, G.; Xiong, J. Stochastic Differential Equations in Infinite-Dimensions: A Brief Survey and Some New Directions of Research. Multivariate Analysis: Future Directions (University Park, PA, 1992); North Holland Ser. Statist. Probab., 5; North-Holland: Amsterdam, 1993; 267-277.
27. Kallianpur, G.; Xiong, J. Asymptotic behaviour of a system of interacting nuclear-space-valued stochastic differential equations driven by Poisson random measures. Appl. Math. Ptim. 1994, 30 (2), 175-201.
28. Kallianpur, O.; Xiong, J. Stochastic Differential Equations in Infinite-Dimensional Spaces; Institute of Mathematical Statistics Lecture Notes - Monograph Series, 26; Institute of Mathematical Statistics: Hayward, CA, 1995.
29. Kallianpur, G.; Xiong, J.; Hardy, G.; Ramasubramanian, S. The existence and uniqueness of solutions of nuclear space-valued stochastic differential equations driven by Poisson random measures. Stochastics 1994, 50, 85-122.
30. Krakowiak, W. Operator-stable probability measures on Banach spaces. Coll. Math. 1979, 41 (2), 313-326.
31. Kunita, H. Stochastic integrals based on martingales taking values in Hubert spaces. Nagoya Math. J. 1970, 38, 41-52.
32. Kwapién, S. Isomorphic characterisations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. 1972, 44, 583-595.
33. Kwapién, S.; Woyczynski, W.A. Random series and stochastic integrals: Simple and multiple. In Probability and its Applications; Birkhäuser Inc.: Boston, 1992.
34. Lévy, P. Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa (2) 1934, 3, 337-366; 4, 217-218. (Reprinted in Oeuvre de Paul Lévy, Vol. 4, Gauthier-Villars, Paris, 1980.)
35. Lévy, P. Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa (2) 1934, 3, 337-366; 4, 217-218. (Reprinted in Oeuvre de Paul Lévy, Vol. 4, Gauthier-Villars, Paris, 1980.)
36. Lévy, P. Sur les intégrales dont les éléments sont des variables aléatoires indépendantes, Ann. Scuola Norm. Sup. Pisa (2) 1934, 3, 337-366; 4, 217-218. (Reprinted in Oeuvre de Paul Lévy, Vol. 4, Gauthier-Villars, Paris, 1980.)
37. Levy, P. Théorie de l'addition de Variables Aléatoires; Gauthier-Villars: Paris, 1937.
38. Linde, W. Infinitely Divisible and Stable Measures on Banach spaces. Teubner-Texte zur Mathematik, Band 58, Leipzig, 1983.
39. Le Page, R. Multidimensional Infinitely Divisibles Variables and Processes II. Probability in Banach spaces III (Medford, MA., 1980); Lect. N. Maths, 860; Springer: Berlin, New York, 1981; 279-284.
40. Mandrekar, V. Characterisation of Banach spaces through validity of Bochner theorem. In Vector Space Measures and Applications I; Proc. Dublin 1977; Lect. N. Maths. 644, Arou, R.M., Dirren, S., Eds.; Springer Verlag: Berlin, 1978; 314-326.
41. Mandrekar, V.; Rüdiger, B. Existence and uniqueness of path wise solutions for stochastic integral equations driven by non Gaussian noise on separable Banach spaces.
42. Métivier, M. Semimartingales, A Course on Stochastic Processes; De Gruyter Studies in Mathematics 2; Walter de Gruyter: Berlin, New York, 1982.
43. Mikulevicius, R.; Rozovskii, B.L. Normalized stochastic integrals in topological vector spaces. In Séminaire de Probabilités XXXII; Lecture Notes in Mathematics 1686; Azéma, J., Emery, M., Ledoux, M., Yor, M., Eds.; Springer Verlag: Berlin, 1998.
44. Parthasarathy, K.R. Probability Measure on Metric Spaces; Academic Press: New York, 1967.
45. Pratelli, M. Intégration Stochastique et Géométrie des Espaces de Banach; Séminaire de Probabilités, XXII; Lecture Notes in Math. 1321; Springer: Berlin, 1988; 129-137.
46. Privault, N.; Wu, J.L. Poisson stochastic integration on Hubert spaces. Ann. Math. Biaise Pascal 1999, 6 (2), 41-60.
47. Protter, P. Stochastic Integration and Differential Equations, A New Approach; Applications of Mathematics 21; Springer Verlag: Berlin, Heidelberg, New York, 1995. Third printing.
48. Rosinski, J. On stochastic integral representation of stable processes with sample paths in Banach spaces. J. Multiv. Anal. 1986, 20, 277-302.
49. Rosinski, J. Random integrals of space valued functions. Stud. Math. 1984, LXXVII.
50. Rosinski, J. Bilinear random integrals. Diss. Math. 1987, 259, 71.
51. Rüdiger, B. Stochastic integration w.r.t. compensated Poisson random measures on separable Banach spaces. Stoch. Stoch. Reports 2004, 76, 273-292.
52. Rüdiger, B. Ziglio, G. The Ito formula for Banach valued stochastic integrals obtained by integration with respect to compensated Poisson random measures, SFB 611 Bonn, Reprint 187, 2004, Submitted.
53. Sato, K. Lévy Processes and Infinitely Divisible Distributions; Cambridge University Press: Cambridge, U.K., 1999.
54. Schäfer, J. Multiplicative measures, generated Markov semigroups and group valued stochastic processes and fields. J. F. A. 1988, 78, 154-184.
55. Saint Loubert Bié, E. Etude d'une EDPS conduite par un bruit poissonnien. Probab. Theo. reit. Fields 1998, 111, 287-321.
56. Skorohod, A.V. Studies in the Theory of Random Processes; Addison-Wesley Publishing Company, Inc. Reading: Massachusetts, 1965. Originally published as ISSLEDOVANIYA PO TEORII SLUCHAYNYKH PROTSESSOV, Kiev University Press (1961).
57. Tortrat, A. Lois de probabilité sur un espace topologique complètement régulier et produits infinis à termes indépendants dans un groupe topologique. Ann. Inst. Henri Poincaré 1965, 1, 217-237.
58. Üstünel, A.S. Additive processes on nuclear spaces. Ann. Probab. 1984, 12 (3), 858-868.
59. Üstünel, A.S. Stochastic integration on nuclear spaces and its applications. Ann. Inst. Henri Poincaré 1982, XVIII (2) Sect. B., 165-200.
60. Walsh, J.B. An introduction to stochastic partial differential equations. In École dÉté de Probabilités de Saint-Flour XIV-1984; Hennequin, P.L., Ed.; LNM 1180; Springer Verlag: Berlin, Heidelberg, New York, Tokyo, 1986; 266-439.
61. Woyczynski, W.A. Geometry and martingales in Banach spaces, II independent increments. In Advances in Probability 4; M. Dekker: New York, 1978; 267-517.
62. Yor, M. Sur les intégrales stochastiques à valeurs dans un espace de Banach. C.R. Acad. Sci. Paris (A) 1973, 277, 467-469.
63. Yosida, K. Functional Analysis. Classics in Mathematics, 6th Ed.; Springer Verlag: Berlin, 1980.