This is the general form of [[transition-rate-matrices|transition matrices]] of finite-state continuous time Markov chains.
See [[brownian-motion-an-introduction-to-stochastic-processes-schilling-partzsch-pdf]] for further details as detailed in the post https://math.stackexchange.com/questions/694227/what-is-the-importance-of-the-infinitesimal-generator-of-brownian-motion
0.1 General Description
In mathematics - specifically stochastic analysis - the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes significant information about the process.
The generator is used in evolution equations such as the [[kolmogorov-backward-equation]] which describes the evolution of statistics of the process; its \(L^2\) Hermitian adjoint is used in evolution equations such as the Fokker-Planck Equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process.
0.2 Mathematical Description
For a Feller process \((X_{t})_{t\geq 0}\) with Feller semigroup \(T=(T_{t})_{{t\geq 0}}\) and state space \(E\) we define the generator \((A,D(A))\) by \[ \begin{cases} D(A)=\left\{ f\in C_{0}(E):\lim_{ t \downarrow 0 } \frac{T_{t}f-f}{t} \text{ exists as uniform limit} \right\} \\ Af=\lim_{ t \downarrow 0 }\frac{{T_{t}f-f}}{t}, \text{ for any }f\in D(A). \end{cases} \] Here \(C_{0}(E)\) denotes the [[banarch-space|Banarch space]] of continuous functions on \(E\) vanishing at infinity, equipped with the supremum norm and \[ T_{t}f(x)=\mathbb{E}^xf(X_{t})=\mathbb{E}[f(X_{t})|X_{0}=x]. \] ### Generators of Common Processes The general \(n\)-dimensional diffusion process \[ dX_{t}=\mu(t, X_{t})dt+\sigma(t,X_{t})dW_{t}, \] has generator \[ Af=(\nabla f)^T\mu+tr((\nabla^2f)D), \] where \(D=\frac{1}{2}\sigma \sigma^T\) is the diffusion matrix, \(\nabla^2f\) is the Hessian of the function \(f\), and \(tr\) is the trace of the matrix. Its adjoint operator is \[ A^*f=-\sum_{i}\partial_{i}(f\mu_{i})+\sum_{ij}\partial_{ij}(fD_{ij}). \] The following are commonly used special cases for the general \(n\)-dimensional diffusion process:
- Standard Brownian Motion on \(\mathbb{R}^n\), which satisfies the stochastic differential equation \(dX_{t}=dB_{t}\) has generator \[ \frac{1}{2}\Delta \] where \(\Delta\) denotes the Laplace Operator.
- The two-dimensional process \(Y\) satisfying \[ dY_{t}=\begin{pmatrix}dt\\dB_{t}\end{pmatrix} \] where \(B\) is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator \[ Af(t,x)= \frac{{\partial f}}{\partial t}(t,x)+ \frac{1}{2}\frac{\partial^2f}{\partial x^2}(t,x). \]
- The [[ornstein-uhlenbeck-process]] on \(\mathbb{R}\) has generator \[ Af(x)=\theta(\mu-x)f'(x)+\frac{\sigma^2}{2}f''(x). \] similarly, the graph of the Ornstein-Uhlenbeck process has generator \[ Af(t,x)=\frac{\partial f}{\partial t}(t,x)+\theta(\mu-x)\frac{\partial f}{\partial x}(t,x)+\frac{\sigma^2}{2}\frac{\partial^2f}{\partial x^2}(t,x). \]
- A Geometric Brownian Motion on \(\mathbb{R}\) has generator \[ Af(x)=rxf'(x)+ \frac{1}{2}\alpha^2x^2f''(x). \]