Diffusion process: Difference between revisions

Revision as of 18:30, 26 March 2025

Revision as of 18:30, 26 March 2025

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==Infinitesimal Generator ==

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>

=== Transition Probability Density ===

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 ==