Diffusion process

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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.^[1]

Definition of Diffusion Process

Let $E=\mathbb {R} ^{d}$ be the state space with Borel $\sigma$ -algebra, and let $\Omega =C([0,\infty ),\mathbb {R} ^{d})$ denote the canonical space of continuous paths. A family of probability measures $\mathbb {P} _{a;b}^{\xi ,\tau }$ (for $\tau \geq 0$ , $\xi \in \mathbb {R} ^{d}$ ) solves the diffusion problem for coefficients $a^{ij}(x,t)$ (uniformly continuous) and $b^{i}(x,t)$ (bounded, Borel measurable) if:

For all $\Gamma \subseteq \mathbb {R} ^{d}$ ,

  $\mathbb {P} _{a;b}^{\xi ,\tau }{\big (}\psi \in \Omega :\psi (t)=\xi {\text{ for }}0\leq t\leq \tau {\big )}=1.$

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

  $M_{t}^{[f]}=f(\psi (t),t)-f(\psi (\tau ),\tau -\int _{\tau }^{t}\left(L_{a;b}+{\frac {\partial }{\partial s}}\right)f(\psi (s),s)\,ds$ 
 is a local martingale under  $\mathbb {P} _{a;b}^{\xi ,\tau }$ , where 
  $L_{a;b}f=\sum _{i=1}^{d}b^{i}{\frac {\partial f}{\partial x_{i}}}+{\frac {1}{2}}\sum _{i,j=1}^{d}a^{ij}{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}.$

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

Connection to Stochastic Differential Equations

The ${\mathcal {L}}_{a;b}$ -diffusion solves the SDE: $dX_{t}^{i}=\sum _{k=1}^{d}\sigma _{k}^{i}(X_{t})\,dB_{t}^{k}+b^{i}(X_{t})\,dt,$ where $a^{ij}(x)=\sum _{k=1}^{d}\sigma _{k}^{i}(x)\sigma _{k}^{j}(x)$ . Uniqueness of $\mathbb {P} _{a;b}^{\xi ,\tau }$ holds under Lipschitz continuity of $\sigma$ and $b$ .

References

^ "9. Diffusion processes" (PDF). Retrieved October 10, 2011.

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