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The Japanese Contributions to Martingales

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Abstract

analysis Modern probability theory, as founded and developed by distinguished pioneers such as A. N. Kolmogorov, P. Lévy, N. Wiener, and so on, attracted great interest and attention from Japanese mathematicians, including K. Yosida, S. Kakutani, K. Itô and G. Maruyama, and others. Around 1935, Kiyosi Itô (1915–2008), then a student at the University of Tokyo, found Kolmogorov’s recently published book, “Grundbegriffe der Wahrscheinlichkeitsrechnung” one day in a bookstore. As he often recollected in later years, 2 this fortuitous discovery of Kolmogorov’s book gave him one of his motivations for devoting his future life to the study of probability theory. Although the study of modern probability theory in Japan certainly started before 1940, the war disrupted communications with other advanced countries. Under these circumstances, Itô completed two important contributions ([I 1], [I 2]) that are now considered the origin of Itô’s stochastic analysis or Itô’s stochastic calculus. In the first work, he gave a rigorous proof of what is now called the Lévy-Itô theorem for the structure of sample functions of Lévy processes, through which we have a complete understanding of the Lévy-Khinchin formula for canonical forms of infinitely divisible distributions. In the second work, he developed a complete theory of stochastic differential equations determining sample functions of diffusion processes whose laws are described by Kolmogorov’s differential equations. In this work, he introduced the important notion of a stochastic integral and the basic formula now known as Itô’s formula or Itô’s lemma and thus founde

Year: 2013
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