Diffusion process: Difference between revisions

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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.

== Diffusion Process ==

Let <math>E = \mathbba^{Rij}^d(x,t)</math> be theuniformly statecontinuous space with Borel <math>\sigma</math>-algebra,coefficients and let <math>\Omega = Cb^{i}([0x,\infty), \mathbb{R}^dt)</math> denotebe thebounded, canonicalBorel spacemeasurable ofdrift continuous pathsterms. AThere is a unique family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \geqge 0</math>, <math>\xi \in \mathbb{R}^d</math>) solveson the diffusioncanonical problem for coefficientsspace <math>a^{ij}\Omega = C(x[0,t\infty), \mathbb{R}^d)</math>, (uniformlywith continuous)its andBorel <math>b^i(x,t)\sigma</math> (bounded-algebra, Borel measurable)such ifthat:

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>

== Definition of Diffusion Process==

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:

2. (Local Martingale Property) For allevery <math>\Gammaf \subseteqin C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, the process

<math>\mathbb{P}M_t^{\xi,\tau[f]}_{a;b}\big = f( \psi(t),t) \in \Omega :- f(\psi(t\tau),\tau) =- \xiint_\tau^t \textbigl(L_{ for a;b} 0+ \leq t tfrac{\leq partial}{\taupartial s}\bigbigr) = 1.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>, wherewith <math>M_t^{[f]} = 0</math> for <math>t \le \tau</math>.

TheThis family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion, where <math>\mathcal{L}_{a;b} = L_{a;b} + \frac{\partial}{\partial t}</math> is the time‐dependent infinitesimal generator.

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>

=== Connection to Stochastic Differential Equations ===

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

If <math>(X_t)_{t \ge 0}</math> is an <math>\mathcal{L}_{a;b}</math>-diffusion, it satisfies the SDE <math>dX_t^i = \sqrt{2\nu}\sum_{k=1}^{d}\sigma^i_k(X_t)\,dB_t^k + b^i(X_t)\,dt</math>,

provided <math>a^{ij}(x) = \sum_{k=1}^d \sigma^i_k(x)\sigma^j_k(x)</math>, and <math>\sigma^{ij}(x)</math>, <math>b^i(x)</math> are Lipschitz continuous. By Itô's lemma, for <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math> we have <math>df(X_t,t) = \Bigl(\tfrac{\partial f}{\partial t} + \sum_{i=1}^d b^i \tfrac{\partial f}{\partial x_i} + \nu \sum_{i,j=1}^d a^{ij}\tfrac{\partial^2 f}{\partial x_i \partial x_j}\Bigr)\,dt + \text{(martingale terms)}.</math>

=== Infinitesimal Generator ===

== Connection to Stochastic Differential Equations ==

The infinitesimal generator <math>\mathcal{A}</math> of <math>X_t</math> is defined for <math>f \in C^{2,1}(\mathbb{R}^d \times \mathbb{R}^+)</math> by <math>\mathcal{A}f(\mathbf{x},t) = \sum_{i=1}^d b^i(\mathbf{x},t)\,\tfrac{\partial f}{\partial x_i} + \nu \sum_{i,j=1}^d a^{ij}(\mathbf{x},t)\,\tfrac{\partial^2 f}{\partial x_i \partial x_j} + \tfrac{\partial f}{\partial t}.</math>

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>

=== Transition Probability Density ===

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>.

For a diffusion process <math>(X_t, \mathbb{P}^{\xi,\tau}_{a;b})</math>, the transition probability function is <math>H_{a;b}(\tau,\xi,t,\mathrm{d}x) = \mathbb{P}^{\xi,\tau}_{a;b}[\psi : \psi(t)\in\mathrm{d}x].</math>

Under uniform ellipticity of <math>a^{ij}(x,t)</math>, this measure has a density <math>h_{a;b}(\tau,\xi,t,x)</math> w.r.t. Lebesgue measure, satisfying <math>\tfrac{\partial h}{\partial t} = \mathcal{A}^* h,</math> where <math>\mathcal{A}^*</math> is the adjoint of the infinitesimal generator.

== See also ==

* [[Stochastic differential equation]]

*[[Diffusion]]

* [[Itô diffusioncalculus]]

* [[Fokker–Planck equation]]

*[[Jump diffusion]]

* [[Sample-continuousMarkov process]]

* [[Diffusion]]

* [[Itô diffusion]]

* [[Jump diffusion]]

* [[Sample-continuous process]]

== References ==