The story of modern finance starts from a faucet. The Navier–Stokes equations is considered to be the first step to understand the elusive phenomenon of turbulence. The Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. There is a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem. Call it poetic justice, but solving finance is a scientific problem connected to turbulence in the faucet at your home.
In 1822 Claude-Louis Navier and later from 1842–1850 George Gabriel Stokes write the Navier–Stokes equations. The equations arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term and a pressure term—hence describing the viscous flow.
In 1880 Thorvald Thiele teaches Neils Bohr mathematics as a Professor of Astronomy in Copenhagen and published an article on time series which effectively creates a model of Brownian motion.
In 1888 Joseph Bertrand writes "Calcul des Probabilit ́es and talks about the reflection principle in the context of gambling losses.
In 1897 Joseph Valentin Boussinesq writes the "Théorie de l'écoulement tourbillonnant et tumultueux des liquides dans les lits rectilignes a grande section" explaining fluid dynamics and the Boussinesq approximation, which is used in the field of buoyancy-driven flow. He makes further contributions across mathematical physics, notably in the understanding of turbulence and the hydrodynamic boundary layer.
In 1900 Louis Bachelier writes "Th ́eorie de la Sp ́eculation". It is from Boussinesq that Bachelier learns the theory of heat and it was on Boussinesq’s work in fluid mechanics that Bachelier’s second ‘thesis’ was based. Bachelier introduces mathematical finance to the world and also provided a kind of agenda for probability theory and stochastic analysis for the next 65 years or so.
In 1905 Albert Einstein in his miraculous year is unaware of Bachelier’s work and introduces his mathematical model of Brownian motion. Einstein’s motivation is quite different from Bachelier’s. Einstein also derives the connection between Brownian motion and the heat equation.
In 1906 A. A. Markov formalizes the Markov chain. Bachelier had used the ‘lack of memory’ property for the price process earlier.
In 1909 Émile Borel 1909 establishes the mathematical theory of probability based on integration and measure. He proves a version of the strong law of large numbers.
In 1923 Norbert Wiener gives the law of the Fourier coefficients quantifying the ‘roughness’ of the path.
In 1931 Andrey Kolmogorov writes “U ̈ber die analytischen Methoden in der Wahrscheinlichkeitsrechnung” and credits Bachelier as being the first to make systematic the study of the case where the transition probability.
In 1940 Joseph Doob gives rigorous proof for the strong Markov property of the Brownian motion.
In 1960 Paul Samuelson receives a postcard from Jimmy savage alerting him to Bachelier’s work. Samuelson commissions it for translation by A. James. And when asked what he learned from Bachelier’s thesis, Samuelson said ‘it was the tools’ – the panoply of mathematical techniques deployed by Bachelier encompassing Brownian motion, martingales, Markov processes, the heat equation.
In 1960 Rudolf E. Kalman introduces the Kalman filter. In engineering, the accent in the space program era was on dynamical systems and a major boost to the study of differential equations with random inputs.
In 1965 Paul Samuelson and Henry McKean collaborate on a paper on option pricing and recruit Robert Merton, who arrived as a graduate student at MIT in 1967 with a background in applied mathematics.
In 1973 Fischer Black, Robert Merton, and Myron Scholes publish the Black and Scholes model.
Summarized from the brilliant paper written by Mark H. A. Davis, Imperial College, Louis Bachelier’s “Theory of Speculation”.
Mark Herbert Ainsworth Davis (1945–2020) was Professor of Mathematics at Imperial College London. He made fundamental contributions to the theory of stochastic processes, stochastic control and mathematical finance.