Geometric Brownian motion Geometric Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model. A stochastic process "S" is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): where formula_2 is a Wiener process or Brownian motion, and formula_3 ('the
Brownian motion which differs from Smoluchowski by 27/64 can only be put in doubt." Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. If the probability of "m" gains and "n" − "m" losses follows a binomial distribution, with equal "a priori" probabilities of 1/2, the mean total gain is If "n" is large enough so that Stirling's approximation can be used in the form then the expected total gain will be showing that it increases as the