erations research March-Aprll 1959 BROWNIAn MOTION IN THE STOCK MARKETt M. F. M. Osborne U8 Naval Research Laboratory, Washington 26, D C (ReceIved February 6, 1958) that common- stock pnces, and the value of money can be re- garded as an ensemble statistical equilbrium, with properti of partie Y=log P(t+)/P()l, where P(t+r)and Po(t)are the price of the same random choice stock at random times t+r and t, then the steady state dis- tribution function of Y 18 (Y)=exp(y/2o)/v2no't, which 1s pr sely the probability distribution for a particle in Brownian motion the dispersion developed at the end of unit time A similar distribution holds for the value of money, measured approxmately by stock-market Indices Sufficent, but not necessary conditions to derive this distribu tion quantitatively are given by the conditions of trading, and the Weber Fechner law A consequence of the distribution function is that the ex ectation values for prce itself &(P)=fo Pp(r)(dr/dp)dP increase actuation, or dispersion, of P This secular increase has noth ine to do wtn long-term infiation, or the growth of assets n economy, since the expected reciprocal of price, or number of shares pur chasuble in the future per dollar, Increases with r in an identical fashion T IS THE PURPOSE of this paper to show that the logarithms of com- mon-stock prices can be regarded as an ensemble of decisions in a statisti cal steady state, and that thus ensemble of logarithms of prices, each varying wIth the time has a close analogy with the ensemble of coordinates of a large number of molecules We wish to show that the methods of statisti mechanics, normally applied to the latter problem, may also be appled to the former Although the results of this paper were first reached inductively from direct examination of the data on prices, for the sake of clarty we shall present them, at least In part, n a deductive fashion, and compare the t Read before the US Naval Research Laboratory Sold State uary28,1958 145