Geometric Brownian motion by a different Brownian Motion—the model is called a stochastic volatility model. 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):
Reflected Brownian motion "roughly speaking, "Z" is pushed in direction "R" whenever the boundary surface formula_5 is hit, where "R" is the "j"th column of the matrix "R"." Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where "R" is an M-matrix then necessary and sufficient conditions for stability are The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift "μ " and variance "σ" is for