Diffusion process

A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion. The position of the particle is then random; its probability density function as a function of space and time is governed by a convection–diffusion equation.

Mathematical definition

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation.

A diffusion process is defined by the following properties. Let ${\displaystyle a^{ij}(x,t)}$ be uniformly continuous coefficients and ${\displaystyle b^i(x,t)}$ be bounded, Borel measurable drift terms. There is a unique family of probability measures ${\displaystyle {\mathbb{P}}_{a;b}^{\xi ,\tau }}$ (for ${\displaystyle \tau \ge 0}$, ${\displaystyle \xi \in {\mathbb{R}}^d}$) on the canonical space ${\displaystyle \mathrm{\Omega }=C([0,\mathrm{\infty }),{\mathbb{R}}^d)}$, with its Borel ${\displaystyle \sigma }$-algebra, such that:

1. (Initial Condition) The process starts at ${\displaystyle \xi }$ at time ${\displaystyle \tau }$: ${\displaystyle {\mathbb{P}}_{a;b}^{\xi ,\tau }[\psi \in \mathrm{\Omega }:\psi (t)=\xi \text{ for }0\le t\le \tau ]=1.}$

2. (Local Martingale Property) For every ${\displaystyle f\in C^{2,1}({\mathbb{R}}^d\times [\tau ,\mathrm{\infty }))}$, the process

${\displaystyle M_t^{[f]}=f(\psi (t),t)-f(\psi (\tau ),\tau )-{\int }_{\tau }^t(L_{a;b}+\frac{\mathrm{\partial }}{\mathrm{\partial }s})f(\psi (s),s)ds}$ is a local martingale under ${\displaystyle {\mathbb{P}}_{a;b}^{\xi ,\tau }}$ for ${\displaystyle t\ge \tau }$, with ${\displaystyle M_t^{[f]}=0}$ for ${\displaystyle t\le \tau }$.

This family ${\displaystyle {\mathbb{P}}_{a;b}^{\xi ,\tau }}$ is called the ${\displaystyle {\mathcal{L}}_{a;b}}$-diffusion.