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== 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>
== Diffusion Process ==
=== Definition of Diffusion Process (a; b) ===
Let <math>E = \mathbb{R}^d</math> be the state space with Borel <math>\sigma</math>-algebra, and let <math>\Omega = C([0,\infty), \mathbb{R}^d)</math> denote the canonical space of continuous paths. A family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \geq 0</math>, <math>\xi \in \mathbb{R}^d</math>) solves the diffusion problem for coefficients <math>a^{ij}(x,t)</math> (uniformly continuous) and <math>b^i(x,t)</math> (bounded, Borel measurable) if:
* For all <math>\Gamma \subseteq \mathbb{R}^d</math>,
<math>\mathbb{P}^{\xi,\tau}_{a;b}\big( \psi \in \Omega : \psi(t) = \xi \text{ for } 0 \leq t \leq \tau \big) = 1.</math>
* 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 \left( L_{a;b} + \frac{\partial}{\partial s} \right) f(\psi(s), s) \, ds</math>
is a local martingale under <math>\mathbb{P}^{\xi,\tau}_{a;b}</math>, where
<math>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}.</math>
The family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion.
=== Connection to Stochastic Differential Equations ===
The <math>\mathcal{L}_{a;b}</math>-diffusion solves the SDE:
<math>dX_t^i = \sum_{k=1}^d \sigma_k^i(X_t) \, dB_t^k + b^i(X_t) \, dt,</math>
where <math>a^{ij}(x) = \sum_{k=1}^d \sigma_k^i(x)\sigma_k^j(x)</math>. Uniqueness of <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> holds under Lipschitz continuity of <math>\sigma</math> and <math>b</math>.
== See also ==
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