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
Fractional Brownian motion Suppose we want to simulate the values of the "fBM" at times formula_12 using the Cholesky decomposition method. In order to compute formula_15, we can use for instance the Cholesky decomposition method. An alternative method uses the eigenvalues of formula_16: Note that the result is real-valued because formula_36. Note that since the eigenvectors are linearly independent, the matrix formula_39 is invertible. It is also known that where "B" is a standard Brownian motion and Where formula_47 is the Euler hypergeometric integral. Say we want to simulate an "fBm" at points formula_48. The integral may be efficiently computed by Gaussian quadrature.