by Nikolai V. Shokhirev
Up: Stochastic Labs
- Pseudorandom numbers
- Probability transformation
- Diffusion, Wiener process
- Geometric Brownian motion
- Ornstein-Uhlenbeck process
This is a one-dimensional (say, x) continuous process. For any two times t > s it has independent increments W_{t } - W_{s } distributed Normally with mean 0 and variance (t - s).
The above verbal definition can be presented in the following compact form
dx = dW_{t}
Consider this a s a formal definition because x(t) is not differentiable and cannot be governed by any differential equation.
The simplest generalization of the Wiener process is the diffusion with drift where mean is linear in time with some factor μ and its variance is σ^{2} (t - s).
dx = μ t + σ dW_{t}
The above SDE and the corresponding solution techniques (Itô calculus or Stratonovich calculus) are the ways to calculate mean (observable) values. The alternative way is the solution of the following partial differential (PDE) equation for the probability density function ρ(x, t):
This equation is also known as Kolmogorov forward equation. The solution for ρ(x, 0; x_{0}) = δ(x - x_{0}) is
Here t_{0} = 0.
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©Nikolai V. Shokhirev, 2007-2008