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Martingales in Japan

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The Splendors and Miseries of Martingales

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

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Abstract

Japanese mathematicians did not contribute directly to martingale theory before 1960, but many later contributions were based on the stochastic calculus that Kiyosi Itô first introduced in 1942. Itô’s collaboration with Henry McKean on the pathwise construction of diffusions attracted wide interest from students in Japan. Subsequent Japanese contributions in the 1960s included adaptations of results on Markov processes to martingales, such as Itô and Watanabe’s multiplicative analog of the Doob–Meyer decomposition, which involved the introduction of local martingales, contributions to stochastic integration for square-integrable martingales and semimartingales, and contributions to the representation of martingales. Japanese contributions after 1970 included Itô’s reformulation of the stochastic calculus in term of stochastic differentials, Itô’s circle operation, the Itô–Tanaka formula, and the Fukushima decomposition.

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Notes

  1. 1.

    The situation at the time was similar in all work on Markov processes, perhaps with the exception of Doob’s work. The same objects were given different names in each theory; for example, stopping times were called Markov times in Markov process theory. Since then, the two theories have gradually mixed together, bringing remarkable progress to each.

  2. 2.

    Cf. Introduction of [5], which was originally Dynkin’s plenary lecture at ICM 1962, Stockholm.

  3. 3.

    Cf. e.g., [38, p. 109].

  4. 4.

    For an AF, it is equivalent that it have zero expectation and that it be a martingale with respect to the natural filtration of the process.

  5. 5.

    The \(f_i\) are Borel functions on \(\textbf{R}^d\) with certain integrability conditions.

  6. 6.

    A standard terminology now is predictable quadratic co-variation of M and N. Meyer [35] introduced another random inner product [M, N], called the quadratic co-variation of M and N, which plays important role in the study of discontinuous semimartingales.

  7. 7.

    Cf. e.g., [2, 22, 38] as important texts treating the theory.

  8. 8.

    The term semimartingale and its notion were introduced by Meyer [34, 35]. The word was used differently in Doob’s book [4]; there semi-martingale was used to mean submartingale, and lower semi-martingale to mean supermartingale.

  9. 9.

    The notion of the compensator for a point process is key in the martingale theoretic approach to point processes. Indeed, it has much to do with semimartingale theory; the discontinuities of a semimartingale define a point process on the real line and, conversely, a point process on a general state space defines a discontinuous semimartingale by a projection of the state space to the real line. Cf. e.g. [9, 22, 25], for the martingale-theoretic approach to point processes and applications.

  10. 10.

    We state it in the one-dimensional case; its multi-dimensional extension is straightforward.

  11. 11.

    Cf. D. W. Stroock’s interesting remark [42, p. 180].

References

  1. Davis, M.H.A., Varaiya, P.: The multiplicity of an increasing family of \(\sigma \)-fields. The Annals of Probability 2(5), 958–963 (1974)

    Article  MathSciNet  Google Scholar 

  2. Dellacherie, C., Meyer, P.A.: Probabilities and Potential B: Theory of Martingales. North-Holland, Amsterdam (1982). Chapters V–VIII

    Google Scholar 

  3. Doléans-Dade, C.: Quelques applications de la formule de changement de variables pour les semimartingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 16(3), 181–194 (1970)

    Article  MathSciNet  Google Scholar 

  4. Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

    MATH  Google Scholar 

  5. Dynkin, E.B.: Markov Processes. Springer (1965)

    Google Scholar 

  6. Fisk, D.L.: Quasi-martingales. Transactions of the American Mathematical Society 120(3), 369–389 (1965)

    Article  MathSciNet  Google Scholar 

  7. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. de Gruyter, Berlin (1994)

    Book  Google Scholar 

  8. Girsanov, I.V.: On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory of Probability & Its Applications 5(3), 285–301 (1960)

    Article  MathSciNet  Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic Differential equations and Diffusion Processes. North-Holland (1988)

    Google Scholar 

  10. Itô, K.: On stochastic processes (infinitely divisible laws of probability). Japanese Journal of Mathematics 18, 261–301 (1942)

    Article  MathSciNet  Google Scholar 

  11. Itô, K.: Zenkoku sizyo sugaku danwakai-si. Journal Pan-Japan Mathematical Colloquium (1077), 1352–1400 (1942). Translated as “Differential equations determining a Markoff process” on pp. 42–75 of [43]

    Google Scholar 

  12. Ito, K.: Foundations of Probability Theory (in Japanese). Iwanami (1944)

    Google Scholar 

  13. Itô, K.: Multiple Wiener integral. Journal of the Mathematical Society of Japan 3(1), 157–169 (1951)

    Article  MathSciNet  Google Scholar 

  14. Itô, K.: On stochastic differential equations. Memoirs of the American Mathematical Society (4), 51 pp. (1951)

    Google Scholar 

  15. Ito, K.: Probability Theory (in Japanese). Iwanami (1953)

    Google Scholar 

  16. Ito, K.: Stochastic differentials. Applied Mathematics and Optimization 1(4), 374–381 (1975)

    Article  MathSciNet  Google Scholar 

  17. Itô, K.: Extension of stochastic integrals. In: Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto,1976, pp. 95–109. Mathematical Society of Japan (1978)

    Google Scholar 

  18. Itô, K.: A note of thanks by K. Itô. In: Stochastic Analysis and Related Topics in Kyoto: In honour of Kiyosi Itô. Mathematical Society of Japan (2004)

    Google Scholar 

  19. Itô, K.: Memoirs of my research on stochastic analysis. In: Stochastic Analysis and Applications, pp. 1–5. Springer (2007)

    Google Scholar 

  20. Itô, K., McKean Jr., H.P.: Diffusion Processes and their Sample Paths. Springer (1965)

    Google Scholar 

  21. Ito, K., Watanabe, S.: Transformation of Markov processes by multiplicative functionals. Annales de l’institut Fourier 15(1), 13–30 (1965)

    Article  MathSciNet  Google Scholar 

  22. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer (1987)

    Google Scholar 

  23. Jeulin, T., Yor, M.: Grossissements de filtrations: exemples et applications. Springer (1985)

    Google Scholar 

  24. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer (1988)

    Google Scholar 

  25. Kasahara, Y., Watanabe, S.: Limit theorems for point processes and their functionals. Journal of the Mathematical Society of Japan 38(3), 543–574 (1986)

    Article  MathSciNet  Google Scholar 

  26. Khinchin, A.Y.: On the basic theorems of information theory. Uspekhi matematicheskikh nauk 11(1), 17–75 (1956)

    MathSciNet  MATH  Google Scholar 

  27. Kunita, H., Watanabe, S.: On square-integrable martingales. Nagoya Journal of Mathematics 30, 209–245 (1967)

    Article  MathSciNet  Google Scholar 

  28. Le Gall, J.F.: Applications du temps local aux équations différentielles stochastiques unidimensionnelles. In: Séminaire de Probabilités XVII 1981/82, pp. 15–31. Springer (1983)

    Google Scholar 

  29. Maruyama, G.: On the transition probability functions of the Markov process. Natl. Sci. Rep. Ochanomizu Univ 5, 10–20 (1954)

    MathSciNet  MATH  Google Scholar 

  30. McKean, H.P.: Stochastic Integrals. Academic Press (1969)

    Google Scholar 

  31. Meyer, P.A.: A decomposition theorem for supermartingales. Illinois Journal of Mathematics 6(2), 193–205 (1962)

    Article  MathSciNet  Google Scholar 

  32. Meyer, P.A.: Fonctionnelles multiplicatives et additives de Markov. Annales de l’institut Fourier 12, 125–230 (1962)

    Article  Google Scholar 

  33. Meyer, P.A.: Decomposition of supermartingales: the uniqueness theorem. Illinois Journal of Mathematics 7(1), 1–17 (1963)

    Article  MathSciNet  Google Scholar 

  34. Meyer, P.A.: Probability and Potentials. Blaisdell (1966)

    Google Scholar 

  35. Meyer, P.A.: Intégrales stochastiques I–IV. Séminaire de probabilités de Strasbourg 1, 72–162 (1967)

    MATH  Google Scholar 

  36. Motoo, M.: Diffusion process corresponding to \(1/2\sum \partial ^2/\partial x^{i2} + \sum b^i(x)\partial /\partial x^i\). Annals of the Institute of Statistical Mathematics 12(1), 37–61 (1960)

    Article  MathSciNet  Google Scholar 

  37. Motoo, M., Watanabe, S.: On a class of additive functionals of Markov processes. Journal of Mathematics of Kyoto University 4(3), 429–469 (1965)

    MathSciNet  MATH  Google Scholar 

  38. Protter, P.: Stochastic Integration and Differential Equations: A New Approach. Springer (1990)

    Google Scholar 

  39. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 2nd edn. Springer (1991)

    Google Scholar 

  40. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 2: Itô Calculus. Wiley (1987)

    Google Scholar 

  41. Stratonovich, R.L.: A new representation for stochastic integrals and equations. SIAM Journal on Control 4(2), 362–371 (1966). Translated from Russian article that appeared in Vestnik Moscow Univ. Ser. I Mat. Mech. 1m 3–12 in 1964

    Google Scholar 

  42. Stroock, D.W.: Markov Processes from K. Itô’s Perspective. Princeton University Press (2003)

    MATH  Google Scholar 

  43. Stroock, D.W., Varadhan, S.R.S. (eds.): Kiyosi Itô: Selected Papers. Springer, New York (1987)

    Google Scholar 

  44. Trotter, H.F.: A property of Brownian motion paths. Illinois Journal of Mathematics 2(3), 425–433 (1958)

    Article  MathSciNet  Google Scholar 

  45. Venttsel’, A.D.: Additive functionals on a multi-dimensional Wiener process. Doklady Akademii Nauk 139(1), 13–16 (1961)

    MathSciNet  Google Scholar 

  46. Watanabe, S.: On discontinuous additive functionals and Lévy measures of a Markov process. Japanese Journal of mathematics 34, 53–70 (1964)

    Article  MathSciNet  Google Scholar 

  47. Williams, D.: Probability with Martingales. Cambridge University Press (1991)

    Google Scholar 

  48. Yamada, T.: Principal values of Brownian local times and their related topics. In: N. Ikeda, et al. (eds.) Itô’s Stochastic Calculus and Probability Theory, pp. 413–422. Springer (1996)

    Google Scholar 

  49. Yor, M.: Some Aspects of Brownian Motion: Part II, Some Recent Martingale Problems. Birkhäuser, Basel (1997)

    Book  Google Scholar 

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Authors and Affiliations

  1. Kyoto University, Kyoto, Japan

    Shinzo Watanabe

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  1. Shinzo Watanabe
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Correspondence to Shinzo Watanabe .

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Editors and Affiliations

  1. LPSM, Sorbonne Université, Paris, France

    Laurent Mazliak

  2. Rutgers Business School, Newark, NJ, USA

    Glenn Shafer

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Watanabe, S. (2022). Martingales in Japan. In: Mazliak, L., Shafer, G. (eds) The Splendors and Miseries of Martingales. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-05988-9_9

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