Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 45–53 | Cite as

On the works of Kiyosi Itô and stochastic analysis

Special Feature: Award of the 1st Gauss Prize to K. Ito

Abstract.

A common feature that can be consistently found in the works of Professor Kiyosi Itô is a leap from the analysis in distribution family level toward the analysis and synthesis in sample paths level, which has turned analytic descriptions into thoroughly stochastic ones.

Keywords and phrases:

Brownian motion Lévy–Itô decomposition Itô integral Itô formula stochastic differential equation Wiener–Itô decomposition one-dimensional diffusion excursions stochastic geometry stochastic control stochastic finance 

Mathematics Subject Classification (2000):

60-02 

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References

  1. 1.
    L. Bachelier, Théorie de la spéculation, Ann. Sci. école Norm. Sup., 17 (1900), 21–86.MathSciNetGoogle Scholar
  2. 2.
    A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U. S. A., 45 (1959), 208–215.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Economy, 81 (1973), 637–659.CrossRefGoogle Scholar
  4. 4.
    J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc., 42 (1937), 107–140. MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953.MATHGoogle Scholar
  6. 6.
    E. B. Dynkin, Markov Processes, Moscow, 1963; English translation (in two volumes), Springer-Verlag, Berlin, 1965.Google Scholar
  7. 7.
    A. Einstein, Über die von der molekularkinematischen Theorie der Wärme geforderte Bewegung von ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., 17 (1905), 549–560. Google Scholar
  8. 8.
    W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann., 113 (1936), 113–160. MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468–519. CrossRefMathSciNetGoogle Scholar
  10. 10.
    W. Feller, On second order differential operators, Ann. of Math. (2), 61 (1955), 90–105.CrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland; Kodansha, 1980.Google Scholar
  12. 12.
    G. A. Hunt, Markov processes and potentials, I, II, III, Illinois J. Math., 1 (1957), 44–93; 1 (1957), 316–369; 2 (1958), 151–213. Google Scholar
  13. 13.
    N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland; Kodansha, 1980.Google Scholar
  14. 14.
    K. Itô, On stochastic processes. I. (Infinitely divisible laws of probability), doctoral thesis, Jap. J. Math., 18 (1942), 261–301.Google Scholar
  15. 15.
    K. Itô, Differential equations determining a Markoff process (in Japanese), J. Pan-Japan Math. Coll. 1077 (1942), 1352–1400; (in English) In: Kiyosi Itô Selected Papers, Springer-Verlag, 1986, pp. 42–75.Google Scholar
  16. 16.
    K. Itô, A kinematic theory of turbulence, Proc. Imp. Acad. Tokyo, 20 (1944), 120–122. MATHMathSciNetGoogle Scholar
  17. 17.
    K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 4 (1951), 1–51. Google Scholar
  18. 18.
    K. Itô, Multiple Wiener integral, J. Math. Soc. Japan, 3 (1951), 157–169. MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Itô, Stochastic Processes I, II (in Japanese), Iwanami-Shoten, Tokyo, 1957; (English translation by Yuji Ito) Essentials of Stochastic Processes, Transl. Math. Monogr. 231, Amer. Math. Soc., Providence, RI, 2006.Google Scholar
  20. 20.
    K. Itô, Wiener integral and Feynman integral, In: Proc. Fourth Berkeley Sympos. Math. Statist. and Probability, II, 1960, pp. 227–238.Google Scholar
  21. 21.
    K. Itô, Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral, In: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, II, 1965, pp. 145–161.Google Scholar
  22. 22.
    K. Itô, Poisson point processes and their application to Markov processes, Lecture note of Mathematics Department, Kyoto Univ., preprint (1969).Google Scholar
  23. 23.
    K. Itô, Poisson point processes attached to Markov processes, In: Proc. Sixth Berkeley Sympos. Math. Statist. and Probability, III, 1970, pp. 225–239.Google Scholar
  24. 24.
    K. Itô, Stochastic differentials, Appl. Math. Optim., 1 (1974), 374–381.CrossRefMathSciNetGoogle Scholar
  25. 25.
    K. Itô, Extensions of stochastic integrals, In: Proc. Int. Symp. Stochastic Differential Equations, Kyoto, 1976, (ed. K. Itô), Kinokuniya, Tokyo, 1978, pp. 95–109.Google Scholar
  26. 26.
    Kiyosi Itô Selected Papers, (eds. D. W. Stroock and S. R. S. Varadhan), Springer-Verlag, 1986.Google Scholar
  27. 27.
    K. Itô, Memoirs of My Research on Stochastic Analysis, In: Proc. The Abel Symp. 2005, Stochastic Anal. Appl.–A Symposium in Honor of Kiyosi Itô–, Springer, 2007, pp. 1–5.Google Scholar
  28. 28.
    K. Itô and H. P. McKean, Jr., Brownian motions on a half line, Illinois J. Math., 7 (1963), 181–231. MATHMathSciNetGoogle Scholar
  29. 29.
    K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965; In: Classics Math., Springer-Verlag, 1996.Google Scholar
  30. 30.
    K. Itô and S. Watanabe, Introduction to stochastic differential equations, In: Proc. Int. Symp. Stochastic Differential Equations, Kyoto, 1976, (ed. K. Itô), Kinokuniya, Tokyo, 1978, pp. 1–30.Google Scholar
  31. 31.
    S. Kakutani, Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo, 20 (1944), 706–714. MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    A. Kolmogorov, Über die analytischen Methoden in der Wahrscheilichkeitsrechnung, Math. Ann., 104 (1931), 415–458.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Erg. der Math., Berlin, 1933.Google Scholar
  34. 34.
    H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J., 30 (1967), 209–245.MATHMathSciNetGoogle Scholar
  35. 35.
    P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937. Google Scholar
  36. 36.
    P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1948.MATHGoogle Scholar
  37. 37.
    B. Maisonneuve, Exit systems, Ann. Probability, 3 (1975), 399–411.MATHMathSciNetGoogle Scholar
  38. 38.
    H. P. McKean, Jr., Stochastic Integrals, Academic Press, New York and London, 1969. MATHGoogle Scholar
  39. 39.
    R. C. Merton, Theory of rational option pricing, Bell J. Econom. and Management Sci., 4 (1973), 141–183. CrossRefMathSciNetGoogle Scholar
  40. 40.
    R. C. Merton, Continuous-Time Finance, Blackwell, Cambridge, MA, 1990. Google Scholar
  41. 41.
    P. A. Meyer, Intégrals stochastiques (4 exposés), In: Séminaire de Probabilités I, Lecture Notes in Math. 39, Springer-Verlag, 1967, pp. 72–162.Google Scholar
  42. 42.
    M. Motoo and S. Watanabe, On a class of additive functionals of Markov processes, J. Math. Kyoto Univ., 4 (1965), 429–469. MATHMathSciNetGoogle Scholar
  43. 43.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, 1991.Google Scholar
  44. 44.
    L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, two volumes, John Wiley and Sons Ltd., 1979; 2nd Edition, Cambridge University Press, 2000.Google Scholar
  45. 45.
    D. Stroock, Markov Processes from K. Itô’s Perspective, Princeton Univ. Press, 2003.Google Scholar
  46. 46.
    N. Wiener, Differential space, J. Math. Phys., 2 (1923), 131–174.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer 2007

Authors and Affiliations

  1. 1.Osaka UniversityOsakaJapan

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