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Stochastic Integration

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Continuous Martingales and Brownian Motion

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 293))

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Abstract

In this chapter, we introduce some basic techniques and notions which will be used throughout the sequel. Once and for all, we consider below, a filtered probability space (Ω,ℱ,ℱ t , P) and we suppose that each ℱ t contains all the sets of P-measure zero in ℱ. As a result, any limit (almost-sure, in the mean, etc...) of adapted processes is an adapted process; a process which is indistinguishable from an adapted process is adapted.

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Notes and Comments

  1. Ikeda, N., and Watanabe, S. comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14 (1977) 619 - 633

    MathSciNet  MATH  Google Scholar 

  2. Meyer, P.A. Processus de Markov. Lecture Notes in Mathematics, vol. 26, Springer, Berlin Heidelberg New York 1967

    Google Scholar 

  3. Métivier, M. Semimartingales: a course on stochastic processes. de Gruyter, Berlin New York 1982

    Google Scholar 

  4. Fujisaki, M., Kallianpur, G., and Kunita, H. ochastic differential equations for the non-linear filtering problem. Osaka J. Math. 9 (1972) 19 - 40

    MathSciNet  MATH  Google Scholar 

  5. Getoor, R.K. other limit theorem for local time. Z.W. 34 (1976) 1 - 10

    Article  Google Scholar 

  6. Jacod, J., and Yor, M. Etude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales. Z.W. 38 (1977) 83 - 125

    Article  MathSciNet  MATH  Google Scholar 

  7. Emery, M. ne définition faible de BMO. Ann. I.H.P. 21 (1) (1985) 59 - 71

    MathSciNet  MATH  Google Scholar 

  8. Getoor, R.K., and Sharpe, M.J. onformal martingales. Invent. Math. 16 (1972) 271 - 308

    Article  MathSciNet  MATH  Google Scholar 

  9. Maisonneuve, B. Systèmes régénératifs. Astérisque 15. Société Mathématique de France 1974

    Google Scholar 

  10. Paley, R., Wiener, N., and Zygmund, A.Note on random functions. Math. Z. 37 (1933) 647 - 668

    Article  MathSciNet  Google Scholar 

  11. Shiga, T., and Watanabe, S. Bessel diffusions as a one-parameter family of diffusion processes. Z.W. 27 (1973) 37 - 46

    Article  MathSciNet  MATH  Google Scholar 

  12. Isaacson, D. tochastic integrals and derivatives. Ann. Math. Stat. 40 (1969) 1610 - 1616

    Article  MATH  Google Scholar 

  13. Yan, J.A., and Yoeurp, C. Représentation des martingales comme intégrales stochastiques de processus optionnels. Sém. Prob. X. Lecture Notes in Mathematics, vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 422 - 431

    Google Scholar 

  14. Yor, M. Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles. Sém. Prob. X. Lecture Notes in Mathematics, vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 481 - 500

    Google Scholar 

  15. Föllmer, H Stochastic holomorphy. Math. Ann. 207 (1974) 245 - 265

    Article  MathSciNet  Google Scholar 

  16. Doléans-Dade, C. Quelques applications de la formule du changement de variables pour les semi-martingales. Z.W. 16 (1970) 181 - 194

    Article  MATH  Google Scholar 

  17. Ruiz de Chavez, J. Le théorème de Paul Lévy pour des mesures signées. Sém. Prob. XVIII. Lecture Notes in Mathematics, vol. 1059. Springer, Berlin Heidelberg New York 1984, pp. 245 - 255

    Google Scholar 

  18. Kallianpur, G. and Robbins, H. Ergodic property of Brownian motion process. Proc. Nat. Acad. Sci. USA 39 (1953) 525 - 533

    Article  MathSciNet  MATH  Google Scholar 

  19. Calais, J.Y., and Génin, M. Sur les martingales locales continues indexées par0, oo L. Sém. Proba. XVII, Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1988, pp. 454 - 466.

    Google Scholar 

  20. Calais, J.Y., and Génin, M. Sur les martingales locales continues indexées par0, oo L. Sém. Proba. XVII, Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1988, pp. 454 - 466.

    Google Scholar 

  21. Carlen, E., and Krée, P. Sharp L°-estimates on multiple stochastic integrals. Ann. Prob. (1991). To appear

    Google Scholar 

  22. Doss, H., and Lenglart, E. ur l’existence, l’unicité et le comportement asymptotique des solutions d’équations différentielles stochastiques. Ann. I.H.P. 14 (1978) 189 - 214

    MathSciNet  MATH  Google Scholar 

  23. Bass, R.F. Joint continuity and representations of additive functionals of d-dimensional Brownian motion. Stoch. Proc. Appl. 17 (1984) 211 - 228

    Article  MathSciNet  MATH  Google Scholar 

  24. Burkholder, D.L. Martingale transforms. Ann. Math. Stat. 37 (1966) 1494 - 1504

    Article  MathSciNet  MATH  Google Scholar 

  25. Chung, K.L. Excursions in Brownian motion. Ark. för Math. 14 (1976) 155 - 177

    Article  MATH  Google Scholar 

  26. Chung, K.L., and Williams, R.T. Introduction to stochastic integration. Birkhäuser, Boston 1983

    Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Paris VII, 2, Place de Jussieu, F-75251, Paris Cedex 05, France

    Daniel Revuz

  2. Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place de Jussieu, Tour 56, F-75252, Paris Cedex 05, France

    Marc Yor

Authors
  1. Daniel Revuz
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  2. Marc Yor
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© 1991 Springer-Verlag Berlin Heidelberg

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Revuz, D., Yor, M. (1991). Stochastic Integration. In: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol 293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21726-9_5

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  • DOI: https://doi.org/10.1007/978-3-662-21726-9_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21728-3

  • Online ISBN: 978-3-662-21726-9

  • eBook Packages: Springer Book Archive

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