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 square root of the total population. Suppose that a Brownian particle of mass "M" is surrounded by lighter particles of mass "m" which are traveling at a speed "u". Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be "mu"/"M". This ratio is of the order of 10 cm/s. But we also have to take into consideration that in a gas there will be more than 10 collisions in a second, and even greater in a liquid where we expect that there will be 10 collision in one second. Some