## On the Theory of Brownian Motion.

Lancaster: American Physical Society, 1930.

First edition, offprint issue, of this important paper which introduced the ‘Ornstein-Uhlenbeck process.’ Although its origins lay in statistical mechanics, this process has found a wide variety of applications, notably in mathematical finance.

The Ornstein-Uhlenbeck process, also known as the ‘Langevin equation’ in the physics literature, was originally used to describe Brownian motion, the random motion of a small particle suspended in a fluid due to the collisions of the fluid molecules with the particle. Brownian motion is most simply modelled by the ‘Wiener process’, but this is unrealistic because it fails to take into account the viscosity of the fluid. The Ornstein-Uhlenbeck process is a modification of the Wiener process, but allowing for a frictional drag on the particle proportional to its velocity, in accordance with Stokes’s law.

Over time, the Ornstein-Uhlenbeck process tends to drift towards its long-term average. Such ‘mean-reverting’ processes are naturally suited to modelling several situations in finance. For example, the Ornstein-Uhlenbeck process is used in the ‘Vasicek model,’ a mathematical model describing the evolution of interest rates. As opposed to stock prices, for example, interest rates cannot rise indefinitely: at very high levels they would hamper economic activity, prompting a decrease in interest rates. Similarly, interest rates cannot decrease below zero. As a result, interest rates tend to move in a limited range, showing a tendency to revert to a long term average value. Similarly, the Ornstein-Uhlenbeck process can be used to model the behaviour of commodity prices, since it embodies the economic argument that when prices are ‘too high’ demand will reduce and supply will increase, producing a counter-balancing effect. When prices are ‘too low’ the opposite will occur, again pushing prices back towards some kind of long term average.

Leonard Ornstein (1880-1941) was a Dutch physicist who studied under Hendrik Antoon Lorenz at the University of Leiden, receiving a PhD on a problem in statistical mechanics. In 1914 Ornstein succeeded Peter Debye in the chair of theoretical physics at the University of Utrecht, and over the next three decades carried out important work in experimental atomic physics. The Dutch-American physicist George Uhlenbeck (1900-1988) studied under Paul Ehrenfest, receiving his PhD in 1927 for work on statistical methods in quantum theory. After spending eight years at the University of Michigan, in 1935 he moved to Utrecht to take up the chair vacated by Hendrik Kramers. Although most of his work was in statistical mechanics, he is perhaps best known today for his discovery in 1925 of electron spin, in collaboration with Samuel Goudsmit, another of Ehrenfest’s students.

Not in OCLC.

Offprint from: Physical Review, vol. 36 (1930), pp. 823–841. Original printed self-wrappers, pp. 823-841 [1:blank], very fine.

Item #3253

Price: \$500.00