Diffusion process: Difference between revisions

Revision as of 18:25, 26 March 2025

Revision as of 18:25, 26 March 2025

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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)<math> - <math>\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}<math> + <math>\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.