Louis Bachelier's Theory of Speculation by Louis Bachelier

Author:Louis Bachelier
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2006-03-18T16:00:00+00:00

In his first paper on stochastic integration in 1944 he made sense of the notion of solution to this equation, that is, he gave a rigorous mathematical meaning to the stochastic integral. He also states what one might call ‘the fundamental theorem of calculus’ for functions of Brownian motion. In his second paper of 1951 he stated and proved what is now known as Itô’s formula, which makes the connection with Kolmogorov’s partial differential equations. Whereas the Wiener integral is the integral of a deterministic function against white noise, time playing no role, time is crucial in the Itô integral and, moreover, the integrand can itself be a random function.

In Doob’s 1953 book, Itô’s stochastic calculus is extended to processes with orthogonal increments and then to processes with conditionally orthogonal increments, that is martingales. However, Doob had to make an assumption. In order to be able to define the stochastic integral with respect to the martingale M, he required the existence of a non-random increasing function F(t) such that is also a martingale. (When M is Brownian motion, the function F(t) = t has this property.) For discrete-parameter martingales, the analogous property follows from the Doob Decomposition Theorem, which allows one to write a submartingale (uniquely) as the sum of a martingale and a process with increasing paths, starting from zero, and with the property that the process at time n is measurable with respect to the sigma-field generated by information available up to time n − 1. The continuous-time version of this result, under a certain uniform integrability assumption, is due to Meyer (1962). Uniqueness of what is now called the Doob–Meyer decomposition came a year later in Meyer (1963). In his first paper Meyer proposes, as an application of the decomposition theorem, an extension of Doob’s stochastic integral. A systematic development of these ideas is provided by Courrège (1963), but it was left to Kunita and Watanabe (1967) to provide the analogue of Itô’s formula for these more general stochastic integrals.11

Up to this point, stochastic integration was intimately bound with the theory of Markov processes. This came about through a measuretheoretic constraint: the underlying filtration of σ-algebras was assumed to be quasi-left continuous. (This means that the process has no fixed points of discontinuity.) In 1970, Doléans-Dade and Meyer removed this hypothesis and stochastic integration became purely a martingale theory (or, more precisely, a semimartingale theory). From the mathematical finance point of view, this was a key ingredient in the fundamental papers of Harrison and Kreps (1979) and Harrison and Pliska (1981).

We should not close this section without mentioning the extraordinary story of Vincent Doeblin, without whose untimely death in World War II the whole story could have unfolded differently.12 Born Wolfgang, he was the second son of Alfred Döblin, author of Berlin Alexanderplatz, who being Jewish and left-wing saw his books publicly burned by the Nazis before fleeing Germany in 1933 after the burning of the Reichstag. He fled first to Zurich and then to Paris, where

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