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A Short History of Stochastic Integration and Mathematical Finance

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A Short History of Stochastic Integration and Mathematical Finance A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes – Monograph Series Vol. 45 (2004) 75–91 c Institute of Mathematical Statistics, 2004 A short history of stochastic integration and mathematical finance: The early years, 1880–1970 Robert Jarrow1 and Philip Protter∗1 Cornell University Abstract: We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduc- tion of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 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 Copen- hagen, who effectively created a model of Brownian motion while studying time series in 1880 [81].2; 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 [64] 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 ∗Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092. 1School of Operations Research and Industrial Engineering, Cornell University, 219 Rhodes Hall, Ithaca, NY 14853, USA. e-mail: [email protected] Keywords and phrases: stochastic integration, semimartingales, martingales, Brownian mo- tion, Markov processes, Black-Scholes, options, warrants, contingent claims, hedging strategies, Bachelier, homogeneous chaos, history of mathematics. AMS 2000 subject classifications: 01A60, 60H05, 60H30, 60G44, 60G35, 60G46, 91B70, 91B28, 91B99, 60J45, 60J55, 60J65. 2This was called to our attention by Ragnar Norberg, whom we thank, and the contributions of Thiele are detailed in a paper of Hald [30]. 75 76 R. Jarrow and P. Protter limits. His thesis was appreciated by his mentor H. Poincar´e, 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 Besan¸con, 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 in- crements, 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 process. However, Einstein was unable to show that the process he proposed actu- ally did exist as a mathematical object. This is understandable, since it was 1905, and the ideas of Borel and Lebesgue constructing measure theory were developed only during the first decade of the twentieth century. In 1913 Daniell’s approach to measure theory (in which integrals are defined before measures) appeared, and it was these ideas, combined with Fourier series, that N. Wiener used in 1923 to construct Brownian motion, justifying after the fact Einstein’s approach. Indeed, Wiener used the ideas of measure theory to construct a measure on the path space of continuous functions, giving the canonical path pro- jection process the distribution of what we now know as Brownian motion. Wiener and others proved many properties of the paths of Brownian motion, an activity that continues to this day. Two key properties relating to stochastic integration are that (1) the paths of Brownian motion have a non zero finite quadratic variation, such that on an interval (s, t), the quadratic variation is (t − s) and (2) the paths of Brownian motion have infinite variation on compact time intervals, almost surely. The second property follows easily from the first. Note that if Einstein were to have assumed rectifiable paths, Wiener’s construction would have essentially proved the impossibility of such a model. In recognition of his work, his construction of Brown- ian motion is often referred to as the Wiener process. Wiener also constructed a multiple integral, but it was not what is known today as the “Multiple Wiener In- tegral”: indeed, it was K. Itˆo, in 1951, when trying to understand Wiener’s papers (not an easy task), who refined and greatly improved Wiener’s ideas [36]. The next step in the groundwork for stochastic integration lay with A. N. Kol- mogorov. The beginnings of the theory of stochastic integration, from the non- finance perspective, were motivated and intertwined with the theory of Markov processes, in which Kolmogorov, of course, played a fundamental role. Indeed, in 1931, two years before his famous book establishing a rigorous mathematical basis for Probability Theory using measure theory, Kolmogorov refers to and briefly ex- plains Bachelier’s construction of Brownian motion ([41], pp. 64, 102–103). It is this paper too in which he develops a large part of his theory of Markov processes. Most significantly, in this paper Kolmogorov showed that continuous Markov processes (diffusions) depend essentially on only two parameters: one for the speed of the drift and the other for the size of the purely random part (the diffusive component). He was then able to relate the probability distributions of the process to the solu- tions of partial differential equations, which he solved, and which are now known as “Kolmogorov’s equations.” Of course, Kolmogorov did not have the Itˆointegral available, and thus he relied on an analysis of the semigroup and its infinitesimal generator, and the resulting partial differential equations.3 3J. L. Doob [17] has complained that the PDE methods of Kolmogorov and Feller used to study Markov processes have often been called “analytic”, whereas the method of stochastic differentials Stochastic integration and mathematical finance the early years, 1880–1970 77 After Kolmogorov we turn to the fascinating and tragic story of Vincent Doe- blin (born Wolfgang D¨oblin) the son of the author Alfred D¨oblin, who wrote Berlin Alexanderplatz for example. The D¨oblin family fled the Nazis from Germany, first to Switzerland, and then to Paris. Wolfgang changed his name to Vincent Doe- blin, and became a French citizen, finishing his schooling there and being quickly recognized as an extraordinary mathematical talent. In the late 1920’s Probability Theory was becoming stylish among mathematicians, especially in the two centers, Moscow and Paris. Doeblin joined the probabilists, working on Markov chains and later Markov processes.4 Doeblin wanted to construct a stochastic process with continuous paths that would be consistent with Kolmogorov’s analytic theory of transition probabilities for Markov processes. He ultimately developed a framework to study them which was prescient in regards to future developments. However Doe- blin was drafted, and he volunteered to go to the front. Before he went he sketched out his ideas and he put this work in the safe of the National Academy of Science of France, to be opened only by him or else after 100 years. As the Maginot line fell, to avoid sharing his ideas with the Nazis Doeblin first burned his notes, and then he took his own life. The academy safe was opened only in May 2000, at the request of his brother, Claude Doeblin. It was only then that the far reaching vision of his work became apparent. In those notes, he utilized the new concept of martingales proposed by J. Ville only in 1939 [84] and understood the importance of studying sample paths, instead of relying exclusively on distributional properties. One idea he had was to run Brownian motion by a random clock: what is known today as a time change. The change of time was then related to the diffusion coefficient, and in this way he was able to give a modern treatment of diffusions decades before it was developed otherwise. 5 We turn now to Kiyosi Itˆo, the father of stochastic integration. We will not attempt to reproduce the beautiful summary of his work and contributions pro- vided in 1987 by S. R. S. Varadhan and D. W. Stroock [83], but instead give a short synopsis of what we think were key moments.6 No doubt an attempt to es- tablish a true stochastic differential to be used in the study of Markov processes was one of Itˆo’s primary motivations for studying stochastic integrals, just as it was D¨oblin’s before him, although of course D¨oblin’s work was secret, hidden away in the safe of the French Academy of Science.
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