Diffusion process

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation. A diffusion process is defined by the following properties. Let a^{ij}(x,t) be uniformly continuous coefficients and b^{i}(x,t) be bounded, Borel measurable drift terms. There is a unique family of probability measures \mathbb{P}^{\xi,\tau}_{a;b} (for \tau \ge 0, \xi \in \mathbb{R}^d) on the canonical space \Omega = C([0,\infty), \mathbb{R}^d), with its Borel \sigma-algebra, such that: 1. (Initial Condition) The process starts at \xi at time \tau: \mathbb{P}^{\xi,\tau}_{a;b}[\psi \in \Omega : \psi(t) = \xi \text{ for } 0 \le t \le \tau] = 1. 2. (Local Martingale Property) For every f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty)), the process M_t^{[f]} = f(\psi(t),t) - f(\psi(\tau),\tau) - \int_\tau^t \bigl(L_{a;b} + \tfrac{\partial}{\partial s}\bigr) f(\psi(s),s)\,ds is a local martingale under \mathbb{P}^{\xi,\tau}_{a;b} for t \ge \tau, with M_t^{[f]} = 0 for t \le \tau. This family \mathbb{P}^{\xi,\tau}_{a;b} is called the \mathcal{L}_{a;b}-diffusion. == SDE Construction and Infinitesimal Generator ==

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