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 excursion Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion formula_5 in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of formula_5. For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII. The Brownian excursion area arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.,, and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that formula_19 has density where formula_21 are the zeros of