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\noindent{\bf Louis BACHELIER}\\
b. 11 March 1870 - d. 28 April 1946
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\noindent{\bf Summary}. Bachelier constructed the first mathematical
theory of Brownian motion,
and obtained numerous results which are both remarkable and
unacknowledged, on the
trajectories of stochastic processes.
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For a contemporary mathematician, Bachelier's story is strikingly
1different. He was initially overlooked and obtained his first
permanent university position at the age of 57, achieving fame only
20 years after his death. Bachelier was born in Le Havre, France, into a
family immersed in banking and business. His father,
from the Bordeaux region, was a wine merchant and acted as
Vice-Consul for Venezuela. His maternal grandfather J.B. Fort-Meu,
founded a banking company serving the district of Le Havre.
Bachelier matriculated at Caen in 1888, but his father's death
forced him to interrupt his studies. He took them up again at the
age of 22, and was successful in obtaining an undistinguished
Bachelor's degree in Paris in October 1895. He then defended his
doctoral thesis, entitled ``Th\'eorie de la sp\'eculation" and written
under
the supervision of Henri Poincar\'e, on 29 March 1900; this was
classified as ``honorable". The aim of the thesis was ``the
application of the calculus of probabilities to stockmarket
operations", and was followed by several further notes, papers and
original contributions on probabilities until 1914. Bachelier
was then mobilised, first as a simple private and later as an officer,
until the end of the World War 1.
Bachelier was unable to obtain a university post after he was
awarded his doctorate, and had to find non-academic work.
Nevertheless, he gave some unpaid lectures at the
Sorbonne on the calculus of probabilities from 1910 to 1914. After
returning from the war, he first completed an assignment for the
Ministry of Labour in Alsace-Lorraine, after which he occupied
various precarious positions in the universities of Besan\c{c}on,
Dijon and Rennes. He was finally tenured at Besan\c{c}on in 1927,
where his teaching was well appreciated by his students. He
retired in 1937 and until his death lived at St. Malo.
Bachelier's thesis contains three different versions of the first
mathematical theory of Brownian motion (five years before
Einstein). In modern terminology, Brownian motion was
characterized as:
\noindent a) a process with independent homogeneous increments
whose paths are continuous,\\
\noindent b) the continuous time process which is the limit of
symmetric random walks, and\\
\noindent c) the Markov process whose forward Kolmogorov equation
is the heat equation.
Bachelier made a detailed study of the sample paths of Brownian motion
thirty years before Paul L\'evy, using the reflection
principle and the strong Markov property. One may interpret this
thesis as the fusion of two seemingly very disparate traditions.
The first, which acted as a guideline for Bachelier, was that of
French mathematical physics in the tradition of J. Fourier, G.
Lam\'e and obviously Poincar\'e. Bachelier drew his analogies,
such as the evolution of probabilities, from their ideas. The
second was the tacit rational models of stockmarket speculators
which were used in a less formalised fashion during the second half of
the 19th century. It is possible that a work of Jules Regnault in
1863 already contained, in a more literary form, the conceptual
setting for the application of probability to
stockmarket operations. In particular, this stated that ``the
standard deviation of a large number of operations, is in direct
proportion to
the square root of time". From the mathematical viewpoint,
Bachelier's entire thesis is essentially correct; however, as the
hypotheses were not always precisely stated, several mathematicians
starting with Gevrey and Paul L\'evy thought it was gravely in error.
The newness of the subject at the beginning of the century, and
these negative evaluations resulted in Bachelier's work being
overlooked, and when read, not understood.
However, not only was his thesis
remarkable, but his later researches were even more so,
though they were
largely ignored despite a certain measure of support from
Poincar\'e.
For example, Bachelier's paper of 1906 provides definitions of the classes
of stochastic processes which appeared later: processes with
independent increments, Markov processes, Ornstein-Uhlenbeck
processes. These definitions appear as consequences of a more
general theory: that of stochastic differential equations, which
Bachelier developed, without all the rigour to which we are now
accustomed, using a vocabulary gleaned from games of
chance.
Two functions play a key role in his paper: the first, called
``esp\'erance relative \`a une partie"
is what we now call the ``drift" of the stochastic differential
equation, while the second called ``fonction d'instabilit\'e
relative \`a une partie"
is the coefficient of diffusion of this equation. Bachelier's
arguments concentrate on paths: his equations define a motion, and
call to mind Langevin (1908) and, later, work of Ito and L\'evy.
In addition, Bachelier introduced an interesting theory of
``inverse probability" and of ``probability of causes", that is of
statistical estimation, which he pursued in a treatise on the
calculus of probabilities published in 1912, which is as clear as it is
remarquable. Bachelier was subjected
to much derision. In a confidential note to the Rector at
Besan\c{c}on dated 26 May 1921, the Director of Higher Education,
providing a confidential evaluation, wrote ``His situation is
certainly precarious. But he owes it to me, despite the contrary
advice of most of the mathematicians. He is not a high flier, and
his work is rather peculiar. But he has served well during the
war, and we had not been sufficiently fair to him. In effect,
he has been placed on trial in your Faculty".
The only scholar who truly recognized the depth of Bachelier's work
was Kolmogorov, in his great paper which is at the basis of modern
stochastic processes theory ``Uber die analytischen Methoden in
der Wahrscheinlichkeitsrechnung" of 1931. But after 1930, only
Kolmogorov was read, and only those
parts of Bachelier's work reproduced in the Soviet author's treatises
became known. As for the stock market speculators, they did not
need Bachelier's results, which were considered too theoretical to
be operationally
useful. It was only in the years 1960-1970 that the merit of this
French mathematician was recognized by probabilists, and later by
financiers, as the new rules for trading options became current.
In the 1990's Bachelier Seminars have been created, as has also an
international Bachelier Society focussed on the mathematics of finance.
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\begin{thebibliography}{3}
\bibitem{1} Bachelier, L. (1900). Th\'eorie de la sp\'eculation,
{\it Annales de l'Ecole Normale Sup\'erieure}, {\bf 17}, 21-86.
\bibitem{2} Bachelier, L. (1906). Th\'eorie des probabilit\'es
continues, {\it Journal de Math\'ematiques Pures et Appliqu\'ees,}{\bf 2,}
259-327.
\bibitem{3} Bachelier, L. (1912). {\it Calcul des probabilit\'es},
Gauthier-Villars, Paris.
\bibitem{4} Mandelbrot, B. (1987). Bachelier, Louis (1870-1946).
{\it New Palgrave Dictionary of Political Economy,} Vol. I, 168-169.
\bibitem{5} Langevin,P. (1908). Sur las th\'eorie du mouvement brownien. {\it
Comptes Rendus de l'Acad\'emie des Sciences,} {\bf 146,} 530-533.
\bibitem{6} Regnault, J. (1863). {\it Calcul des chances et philosophie de
la Bourse}, Mallet-Bachelier, Paris.
\bibitem{7} Zylberberg, A. (1990). {\it L'\'Economie Math\'ematique en
France (1870-1914),} Economica, Paris, especially pp. 148-152.
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\hfill{ L. Carraro and P. Cr\'epel}
\end{thebibliography}
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