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Diffusion process

Solution to a stochastic differential equation

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated October 24, 2025

In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion processes are stochastic in nature and hence are 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.

01Mathematical 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.

02SDE Construction and Infinitesimal Generator

It is clear that if we have an {\mathcal {L}}_{a;b}-diffusion, i.e. (X_{t})_{t\geq 0} on (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},\mathbb {P} _{a;b}^{\xi ,\tau }), then X_{t} satisfies the SDE dX_{t}^{i}={\frac {1}{2}}\,\sum _{k=1}^{d}\sigma _{k}^{i}(X_{t})\,dB_{t}^{k}+b^{i}(X_{t})\,dt. In contrast, one can construct this diffusion from that SDE if a^{ij}(x,t)=\sum _{k}\sigma _{i}^{k}(x,t)\,\sigma _{j}^{k}(x,t) and \sigma ^{ij}(x,t), b^{i}(x,t) are Lipschitz continuous. To see this, let X_{t} solve the SDE starting at X_{\tau }=\xi. For f\in C^{2,1}(\mathbb {R} ^{d}\times [\tau ,\infty )), apply Itô's formula: df(X_{t},t)={\bigl (}{\frac {\partial f}{\partial t}}+\sum _{i=1}^{d}b^{i}{\frac {\partial f}{\partial x_{i}}}+v\sum _{i,j=1}^{d}a^{ij}\,{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}{\bigr )}\,dt+\sum _{i,k=1}^{d}{\frac {\partial f}{\partial x_{i}}}\,\sigma _{k}^{i}\,dB_{t}^{k}. Rearranging gives f(X_{t},t)-f(X_{\tau },\tau )-\int _{\tau }^{t}{\bigl (}{\frac {\partial f}{\partial s}}+L_{a;b}f{\bigr )}\,ds=\int _{\tau }^{t}\sum _{i,k=1}^{d}{\frac {\partial f}{\partial x_{i}}}\,\sigma _{k}^{i}\,dB_{s}^{k}, whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of X_{t} defines \mathbb {P} _{a;b}^{\xi ,\tau } on \Omega =C([0,\infty ),\mathbb {R} ^{d}) with the correct initial condition and local martingale property. Uniqueness follows from the Lipschitz continuity of \sigma \!,\!b. In fact, L_{a;b}+{\tfrac {\partial }{\partial s}} coincides with the infinitesimal generator {\mathcal {A}} of this process. If X_{t} solves the SDE, then for f(\mathbf {x} ,t)\in C^{2}(\mathbb {R} ^{d}\times \mathbb {R} ^{+}), the generator {\mathcal {A}} is {\mathcal {A}}f(\mathbf {x} ,t)=\sum _{i=1}^{d}b_{i}(\mathbf {x} ,t)\,{\frac {\partial f}{\partial x_{i}}}+v\sum _{i,j=1}^{d}a_{ij}(\mathbf {x} ,t)\,{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}+{\frac {\partial f}{\partial t}}.

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This article is adapted from the Wikipedia article “Diffusion process”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as prices of listed company shares, random growth models or physical systems that are subjected to thermal fluctuations.

Itô calculus

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion. It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning.

Fokker-Planck equation

In statistical mechanics and information theory, the Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of the position or velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.