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 all "t" ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for "μ" < 0) when taking t → ∞ an exponential distribution For fixed "t", the distribution of "Z(t)" coincides with the distribution of the running maximum "M(t)" of the Brownian motion, But be aware that the distributions of the processes as a whole are very different. In particular, "M(t)" is increasing in "t", which is not the case for "Z(t)". The heat kernel for reflected Brownian motion at formula_9: formula_10 For the plane above formula_11 The stationary distribution of a reflected Brownian motion