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Added a link to the article "Stochastic".

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{{Short description|Solution to a stochastic differential equation}}
{{for|the marketing term|Diffusion of innovations}}
{{one source |date=March 2024}}
In [[probability theory]] and [[statistics]], '''diffusion processes''' are a class of continuous-time [[Markov process]] with [[almost surely]] [[continuous function|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]], [[Artificial neural network|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 [[Sample-continuous_process|continuous sample paths]] for which the [[Kolmogorov_equations|Kolmogorov forward equation]] is the [[Fokker–Planck equation]].<ref>{{cite web|title=9. Diffusion processes|url=http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf|access-date=October 10, 2011}}</ref>

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

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

2. (Local Martingale Property) For every <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, the process
<math>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</math>
is a local martingale under <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> for <math>t \ge \tau</math>, with <math>M_t^{[f]} = 0</math> for <math>t \le \tau</math>.

This family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion.

== SDE Construction and Infinitesimal Generator ==

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

== See also ==
* [[Stochastic differential equation]]
* [[Itô calculus]]
* [[Fokker–Planck equation]]
* [[Markov process]]
* [[Diffusion]]
* [[Itô diffusion]]
* [[Jump diffusion]]
* [[Sample-continuous process]]

== References ==
{{Reflist}}

{{Stochastic processes|state=collapsed}}
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[[Category:Markov processes]]

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Diffusion process - Revision history

imported>Ferenczy: Added a link to the article "Stochastic".