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This 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.
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>
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