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Date: 2002. 5. 22.

How Stochastic Calculus Depends on "History" 





It is often said that "History" with a big H, and "history" with a little h
have sometimes points in common....
The Colloquium talk shall consist of two parts, the first one concerning the
big H, the second one the little h.
Recently, in December 2000, the contents of a sealed envelope (: "pli
cachete", in French)
sent by W. Doeblin in February
1940 from his battalion on the front line in the Ardennes to the Academie
des Sciences
in Paris, were finally published in a special issue of the Comptes Rendus, and
it turned out, that, even at that time, W. Doeblin was using the concept of
martingale,
which he just learned from Jean Ville in 1939, to describe onedimensional
diffusions. In June 1940, W. Doeblin shot himself to avoid being taken
prisoner by the German army.
The era of Ito's stochastic calculus began from 1942, and History has
not permitted that the two points of view (of Doeblin and Ito) develop in
parallel during
the forties.....
Concerning the little h aspect of stochastic calculus,
this refers to the fact that this calculus is relative not only to a
probability
space, but also to a filtration
, which describes the
evolution
of "history" with time .
Nice processes (: integrators) for stochastic calculus are the
semimartingales,
and it is an interesting question to understand how semimartingales
are being transformed as "history" changes. Many examples, related to
Brownian motion,
Brownian bridges, and their zeros, maxima, etc...., may be developed.
The corresponding formulae are the "change of history" counterparts of the more
classical MaruyamaGirsanovVan SchuppenWong formulae when the reference
probability is
being changed.