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Inprobability theory andstatistics,diffusion processes are a class of continuous-timeMarkov process withalmost surelycontinuous sample paths. Diffusion processes arestochastic in nature and hence are used to model many real-life stochastic systems.Brownian motion,reflected Brownian motion andOrnstein–Uhlenbeck processes are examples of diffusion processes. It is used heavily instatistical physics,statistical analysis,information theory,data science,neural networks,finance andmarketing.

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 calledBrownian motion. The position of the particle is then random; itsprobability density function as afunction of space and time is governed by aconvection–diffusion equation.

Adiffusion process is aMarkov process withcontinuous sample paths for which theKolmogorov forward equation is theFokker–Planck equation.^[1]

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.

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

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

• Markov processes

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