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Girsanov’s Theorem and First Applications

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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 study the effect on the space of continuous semimartingales of an absolutely continuous change of probability measure. The results we describe have far-reaching consequences from the theoretical point of view as is hinted at in Sect. 2; they also permit many explicit computations as is seen in Sect. 3.

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  1. Maruyama, G. On the transition probability functions of Markov processes. Nat. Sci. Rep. Ochanomizu Univ. 5 (1954) 10 - 20

    MathSciNet  MATH  Google Scholar 

  2. Maruyama, G. Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo 10 (1955) 48 - 90

    Article  Google Scholar 

  3. Wong, E., and Zakai, M. The oscillation of stochastic integrals. Z.W. 4 (1965) 103-112

    Google Scholar 

  4. Jacod, J., and Mémin, J. Sur l’intégrabilité uniforme des martingales exponentielles. Z.W. 42 (1978) 175 - 204

    Article  Google Scholar 

  5. Dellacherie, C. Meyer, P.A. and Yor, M. Sur certaines propriétés des espaces H’ et BMO. Sém. Prob. XII. Lecture Notes in Mathematics, vol. 649. Springer, Berlin Heidelberg New York 1978, pp. 98 - 113

    Google Scholar 

  6. Yoeurp, C. Contribution au calcul stochastique. Thèse de doctorat d’état, Université de Paris V I, 1982

    Google Scholar 

  7. Yoeurp, C. Théorème de Girsanov généralisé et grossissement de filtrations. In: Grossissement de filtrations: exemples et applications. Lecture Notes in Mathematics, vol. 1118. Springer, Berlin Heidelberg New York 1985, pp. 172 - 196

    Chapter  Google Scholar 

  8. Kazamaki, N., and Sekiguchi, T. Uniform integrability of continuous exponential martingales. Tohoku Math. J. 35 (1983) 289 - 301

    Article  MathSciNet  MATH  Google Scholar 

  9. Yan, J.A. A propos de l’intégrabilité uniforme des martingales exponentielles. Sém. Prob. XVI. Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 338 - 347

    Google Scholar 

  10. Lépingle, D. and Mémin, J. Sur l’intégrabilité des martingales exponentielles. Z.W. 42 (1978) 175 - 203

    Article  MATH  Google Scholar 

  11. Liptser, R.S., and Shiryaev, A.N. Statistics of random processes I and II. Springer Verlag, Berlin, 1977 and 1978

    Google Scholar 

  12. 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 

  13. Koval’chik, I.M. The Wiener integral. Russ. Math. Surv. 18 (1963) 97 - 135

    Article  MATH  Google Scholar 

  14. Clark, J.M.C. The representation of functionals of Brownian motion by stochastic integrals. Ann. Math. Stat. 41 (1970) 1282 - 1295; 42 (1971) 1778

    Google Scholar 

  15. Williams, D. Markov properties of Brownian local time. Bull. Amer. Math. Soc. 76 (1969) 1035 - 1036

    Google Scholar 

  16. Haussman, U.G. unctionals of Itô processes as stochastic integrals. Siam J. Contr. Opt. 16, 2 (1978) 252 - 269

    Article  Google Scholar 

  17. Bismut, J.M. On the set of zeros of certain semi-martingales. Proc. London Math. Soc. (3) 49 (1984) 73 - 86

    Article  MathSciNet  MATH  Google Scholar 

  18. Friedman, A. tochastic differential equations and applications I and II. Academic Press, New-York, 1975 and 1976

    Google Scholar 

  19. Schilder, M. Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125 (1966) 63 - 85

    Article  Google Scholar 

  20. Stroock, D.W., and Varadhan, S.R.S. Multidimensional diffusion processes. Springer, Berlin Heidelberg New York 1979

    MATH  Google Scholar 

  21. Stroock, D.W. An introduction to the theory of large deviations. (Universitext). Springer, Berlin Heidelberg New York 1984

    Google Scholar 

  22. Strassen, V. An invariance principle for the law of the iterated logarithm. Z.W. 3 (1964) 211 - 226

    Article  MathSciNet  MATH  Google Scholar 

  23. Stroock, D.W. On the growth of stochastic integrals. Z.W. 18 (1971) 340 - 344

    Article  MathSciNet  MATH  Google Scholar 

  24. Choyer, J. On Strassen’s version of the log log law. Z.W. 8 (1967) 83 - 90

    Article  Google Scholar 

  25. Mueller, C. A unification of Strassen’s law and Lévy’s modulus of continuity. Z.W. 56 (1981) 163 - 179

    Article  MathSciNet  MATH  Google Scholar 

  26. Yor, M. Formule de Cauchy relative à certains lacets browniens. Bull. Soc. Math. France 105 (1977) 3 - 31

    MathSciNet  MATH  Google Scholar 

  27. Kunita, H. Some extensions of Itô’s formula. Sém. Prob. XV. Lecture Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 118 - 141

    Google Scholar 

  28. Yor, M. Loi de l’indice du lacet brownien et distribution de Hartman-Watson. Z.W. 53 (1980) 71 - 95

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Nagasawa, M. Transformations of diffusions and Schrödinger processes. Prob. Th. Rel. F. 82 (1989) 106 - 136

    Google Scholar 

  31. Elworthy, K.D tochastic differential equations on manifolds. London Math. Soc. Lecture Notes Series 70

    Google Scholar 

  32. Elworthy, K.D., and Truman, A. he diffusion equation: an elementary formula. In: Stochastic processes in quantum physics (ed. S. Albeverio et al.). Lecture Notes in Physics, vol. 173. Springer, Berlin Heidelberg New York 1982, pp. 136 - 146

    Google Scholar 

  33. Ezawa, H., Klauder, J.R., and Shepp. L.A. estigial effects of singular potentials in diffusion theory and quantum mechanics. J. Math. Phys. 16, 4 (1975) 783 - 799

    Article  MathSciNet  Google Scholar 

  34. Fukushima, M., and Takeda, M. transformation of symmetric Markov processes and the Donsker-Varadhan theory. Osaka J. Math. 21 (1984) 311 - 326

    MathSciNet  MATH  Google Scholar 

  35. Oshima, Y., and Takeda, M. On a transformation of symmetric Markov processes and recurrence property. Lecture Notes in Mathematics, vol. 1250, Proceedings Bielefeld. Springer, Berlin Heidelberg New York 1987

    Google Scholar 

  36. Kunita, H. Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 (1969) 1 - 26

    Google Scholar 

  37. Roth, J.P. Opérateurs dissipatifs et semi-groupes dans les espaces de fonctions continues. Ann. Inst. Fourier 26 (1976) 1 - 97

    Google Scholar 

  38. Mokobodzki, G. Opérateur carré du champ: un contre-exemple. Sém. Prob. XXIII. Lecture Notes in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 324 - 325

    Google Scholar 

  39. Kunita, H. Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 (1969) 1 - 26

    Google Scholar 

  40. Pitman, J.W., and Yor, M. Bessel processes and infinitely divisible laws. In: D. Williams (ed.) Stochastic integrals. Lecture Notes in Mathematics, vol. 851. Springer, Berlin Heidelberg New York 1981

    Google Scholar 

  41. Kent, J. The infinite divisibility of the Von Mises-Fischer distribution for all values of the parameter in all dimensions. Proc. London Math. Soc. 35 (1977) 359 - 384

    MathSciNet  MATH  Google Scholar 

  42. Wendel, J.W. Hitting spheres with Brownian motion. Ann. Prob. 8 (1980) 164 - 169

    Article  MathSciNet  MATH  Google Scholar 

  43. Wendel, J.W. An independence property of Brownian motion with drift. Ann. Prob. 8 (1980) 600 - 601

    Article  Google Scholar 

  44. Pitman, J.W., and Yor, M. A decomposition of Bessel bridges. Z.W. 59 (1982) 425 - 457

    Article  MathSciNet  MATH  Google Scholar 

  45. Pitman, J.W., and Yor, M. Sur une décomposition des ponts de Bessel. In: Functional Analysis in Markov processes. Lecture Notes in Mathematics, vol. 923. Springer, Berlin Heidelberg New York 1982, pp. 276 - 285

    Chapter  Google Scholar 

  46. Gaveau, B. rincipe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta. Math. 139 (1977) 96 - 153

    Article  MathSciNet  Google Scholar 

  47. Bismut, J.M. The Atiyah-Singer theorems. J. Funct. Anal. 57 (1984) 56-99 and 329-348

    Google Scholar 

  48. Bismut, J.M. Formules de localisation et formules de Paul Lévy. Astérisque 157-158, Colloque Paul Lévy sur les processus stochastiques (1988) 37 - 58

    Google Scholar 

  49. Yor, M. A propos de l’inverse du mouvement brownien dans B" (n z 3). Ann. I.H.P. 21, 1 (1985) 27 - 38

    MathSciNet  MATH  Google Scholar 

  50. Schwartz, L. Le mouvement brownien sur WI, en tant que semi-martingale dans SN. Ann. I.H.P. 21, 1 (1985) 15 - 25

    MATH  Google Scholar 

  51. Getoor, R.K. he Brownian escape process. Ann. Prob. 7 (1979) 864 - 867

    Article  MathSciNet  MATH  Google Scholar 

  52. Carne, T.K. Brownian motion and stereographic projection. Ann. I.H.P. 21 (1985) 187 - 196

    MATH  Google Scholar 

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Revuz, D., Yor, M. (1991). Girsanov’s Theorem and First Applications. 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_9

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  • DOI: https://doi.org/10.1007/978-3-662-21726-9_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21728-3

  • Online ISBN: 978-3-662-21726-9

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