Diffusion process

For the marketing term, see Diffusion of innovations.

In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes. It is used heavily in statistical physics, statistical analysis, information theory, data science, neural networks, finance and marketing.

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

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

2. (Local Martingale Property) For every $f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))$, the process $M_{t}^{[f]}=f(\psi (t),t)-f(\psi (\tau ),\tau )-\int _{\tau }^{t}{\bigl (}L_{a;b}+{\tfrac {\partial }{\partial s}}{\bigr )}f(\psi (s),s)\,ds$ is a local martingale under $\mathbb {P} _{a;b}^{\xi ,\tau }$ for $t\geq \tau$, with $M_{t}^{[f]}=0$ for $t\leq \tau$.

This family $\mathbb {P} _{a;b}^{\xi ,\tau }$ is called the ${\mathcal {L}}_{a;b}$-diffusion, where ${\mathcal {L}}_{a;b}=L_{a;b}+{\frac {\partial }{\partial t}}$ is the time‐dependent infinitesimal generator.

Connection to Stochastic Differential Equations

If $(X_{t})_{t\geq 0}$ is an ${\mathcal {L}}_{a;b}$-diffusion, it satisfies the SDE $dX_{t}^{i}={\sqrt {2\nu }}\sum _{k=1}^{d}\sigma _{k}^{i}(X_{t})\,dB_{t}^{k}+b^{i}(X_{t})\,dt$, provided $a^{ij}(x)=\sum _{k=1}^{d}\sigma _{k}^{i}(x)\sigma _{k}^{j}(x)$, and $\sigma ^{ij}(x)$, $b^{i}(x)$ are Lipschitz continuous. By Itô's lemma, for $f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty ))$ we have $df(X_{t},t)={\Bigl (}{\tfrac {\partial f}{\partial t}}+\sum _{i=1}^{d}b^{i}{\tfrac {\partial f}{\partial x_{i}}}+\nu \sum _{i,j=1}^{d}a^{ij}{\tfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}{\Bigr )}\,dt+{\text{(martingale terms)}}.$

Infinitesimal Generator

The infinitesimal generator ${\mathcal {A}}$ of $X_{t}$ is defined for $f\in C^{2,1}(\mathbb {R} ^{d}\times \mathbb {R} ^{+})$ by ${\mathcal {A}}f(\mathbf {x} ,t)=\sum _{i=1}^{d}b^{i}(\mathbf {x} ,t)\,{\tfrac {\partial f}{\partial x_{i}}}+\nu \sum _{i,j=1}^{d}a^{ij}(\mathbf {x} ,t)\,{\tfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}+{\tfrac {\partial f}{\partial t}}.$

Transition Probability Density

For a diffusion process $(X_{t},\mathbb {P} _{a;b}^{\xi ,\tau })$, the transition probability function is $H_{a;b}(\tau ,\xi ,t,\mathrm {d} x)=\mathbb {P} _{a;b}^{\xi ,\tau }[\psi :\psi (t)\in \mathrm {d} x].$ Under uniform ellipticity of $a^{ij}(x,t)$, this measure has a density $h_{a;b}(\tau ,\xi ,t,x)$ w.r.t. Lebesgue measure, satisfying ${\tfrac {\partial h}{\partial t}}={\mathcal {A}}^{*}h,$ where ${\mathcal {A}}^{*}$ is the adjoint of the infinitesimal generator.

See also

• Stochastic differential equation
• Itô calculus
• Fokker–Planck equation
• Markov process
• Diffusion
• Itô diffusion
• Jump diffusion
• Sample-continuous process