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In Stochastic Calculus for Finance II : Continuous-Time Models Steven Shreve writes (p.187):
The modern theory of stochastic calculus developed from the work of Itô [1]. Not only did Itô define the integral with respect to Brownian motion, but he also developed the change-of-variable formula commonly called Itô's rule or Itô's formula. This formula is at the heart of a wide range of useful calculations.

An amazing twist to the story of stochastic calculus has recently emerged. In February 1940, the French National Academy of Sciences received a document from [German-born Jewish mathematician] Wolfgang Doeblin, a French soldier on the German front. Doeblin died shortly thereafter, and the document remained sealed until May 2000. When it was opened, the document was found to contain a construction of the stochastic integral slightly different from Itô's and a clear statement of the change-of variable formula. Doeblin's work [2], Yor's [3] analysis of the work, and a detailed history by Bru [4] of the context of the work appeared in the December 2000 issue of Comptes Rendus de L'Académie des Sciences.
An English translation of this material can be found here. A simple application of the Itô-Döblin formula can be found here. Btw, did you know that the earliest attempt to model Brownian motion mathematically can be traced to T.N. THIELE of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880? Here is the story. In case you do not even know what a Random Walk is, click here or here.

[1] Itô, K. (1944) Stochastic Integral, Proc. Imperial Acad. Tokyo 20, 519-524
[2] Doeblin, W. (1940) Sur l'équation de Kolmogoroff, C. R. Ser. I 331, 1059-1102
[3] Yor, M. (2000) Présentation du pli cacheté, C. R. Ser. I 331, 1033-1036
[4] Bru, B. (200) Un hiver en campagne, C. R. Ser. I 331, 1037-1058

related items: Website der Internationalen Alfred Döblin-Gesellschaft [lang: DE]
isomorphismes (guest) meinte am 5. Dec, 06:02:
cool!