A short history of stochastic integration and mathematical finance the early years, 1880--1970

The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copenhagen, who effectively created a model of Brownian motion while studying time series in 1880 [80].; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [63] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. We go into a little detail about what happened to Bachelier, since he is now seen by many as the founder of modern Mathematical Finance. Ignorant of the work of Thiele (which was little appreciated in its day) and preceding the work of Einstein, Bachelier attempted to model the market noise of the Paris Bourse. Exploiting the ideas of the Central Limit Theorem, and realizing that market noise should be without memory, he reasoned that increments of stock prices should be independent and normally distributed. He combined his reasoning with the Markov property and semigroups, and connected Brownian motion with the heat equation, using that the Gaussian kernel is the fundamental solution to the heat equation. He was able to define other processes related to Brownian motion, such as the maximum change during a time interval (for one dimensional Brownian motion), by using random walks and letting the time steps go to zero, and by then taking limits. His thesis was appreciated by his mentor H. Poincaré, but partially due to the distaste of studying economics as an application of mathematics, he was unable to join the Paris elite, and he spent his career far off in the provincial capital of Besançon, near Switzerland in Eastern France. (More details of this sad story are provided in [11]). Let us now turn to Einstein’s model. In modern terms, Einstein assumed that Brownian motion was a stochastic process with continuous paths, independent increments, and stationary Gaussian increments. He did not assume other reasonable properties (from the standpoint of physics), such as rectifiable paths. If he had assumed this last property, we now know his model would not have existed as a