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== SDE Construction and Infinitesimal Generator ==
It is clear that if we have an <math>\mathcal{L}_{a;b}</math>-diffusion, i.e. <math>(X_t)_{t \ge 0}</math> on <math>(\Omega, \mathcal{F}, \mathcal{F}_t, \mathbb{P}^{\xi,\tau}_{a;b})</math>, then <math>X_t</math> satisfies the SDE <math>dX_t^i = \frac{1}{2}\,\sum_{k=1}^d \sigma^i_k(X_t)\,dB_t^k + b^i(X_t)\,dt</math>. In contrast, one can construct this diffusion from that SDE if <math>a^{ij}(x,t) = \sum_k \sigma^k_i(x,t)\,\sigma^k_j(x,t)</math> and <math>\sigma^{ij}(x,t)</math>, <math>b^i(x,t)</math> are Lipschitz continuous. To see this, let <math>X_t</math> solve the SDE starting at <math>X_\tau = \xi</math>. For <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math>, apply Itô's formula: <math>df(X_t,t) = \bigl(\frac{\partial f}{\partial t} + \sum_{i=1}^d b^i \frac{\partial f}{\partial x_i} + v \sum_{i,j=1}^d a^{ij}\,\frac{\partial^2 f}{\partial x_i \partial x_j}\bigr)\,dt + \sum_{i,k=1}^d \frac{\partial f}{\partial x_i}\,\sigma^i_k\,dB_t^k.</math> Rearranging gives <math>f(X_t,t) - f(X_\tau,\tau) - \int_\tau^t \bigl(\frac{\partial f}{\partial s} + L_{a;b}f\bigr)\,ds = \int_\tau^t \sum_{i,k=1}^d \frac{\partial f}{\partial x_i}\,\sigma^i_k\,dB_s^k,</math> whose right‐hand side is a local martingale, matching the local‐martingale property in the diffusion definition. The law of <math>X_t</math> defines <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> on <math>\Omega = C([0,\infty), \mathbb{R}^d)</math> with the correct initial condition and local martingale property. Uniqueness follows from the Lipschitz continuity of <math>\sigma\!,\!b</math>. In fact, <math>L_{a;b} + \tfrac{\partial}{\partial s}</math> coincides with the infinitesimal generator <math>\mathcal{A}</math> of this process. If <math>X_t</math> solves the SDE, then for <math>f(\mathbf{x},t) \in C^2(\mathbb{R}^d \times \mathbb{R}^+)</math>, the generator <math>\mathcal{A}</math> is <math>\mathcal{A}f(\mathbf{x},t) = \sum_{i=1}^d b_i(\mathbf{x},t)\,\frac{\partial f}{\partial x_i} + v\sum_{i,j=1}^d a_{ij}(\mathbf{x},t)\,\frac{\partial^2 f}{\partial x_i \partial x_j} + \frac{\partial f}{\partial t}.</math>
== See also ==
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