Английская Википедия:Infinitesimal generator (stochastic processes)
Шаблон:About In mathematics — specifically, in 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[1] that encodes a great deal of 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 L2 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.
The Kolmogorov forward equation in the notation is just <math>\partial_t \rho = \mathcal A^* \rho</math>, where <math>\rho</math> is the probability density function, and <math>\mathcal A^*</math> is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.
Definition
General case
For a Feller process <math>(X_t)_{t \geq 0}</math> with Feller semigroup <math>T=(T_t)_{t\geq 0}</math> and state space <math>E</math> we define the generator[1] <math>(A,D(A))</math> by <math display="block">D(A) = \left\{f\in C_0(E): \lim_{t\downarrow 0} \frac{T_t f-f}{t} \text{ exists as uniform limit}\right\},</math> <math display="block">A f = \lim_{t \downarrow 0} \frac{T_t f-f}{t} , ~~ \text{ for any } f\in D(A).</math> Here <math>C_{0}(E)</math> denotes the Banach space of continuous functions on <math>E</math> vanishing at infinity, equipped with the supremum norm, and <math>T_t f(x) = \mathbb{E}^x f(X_t)=\mathbb{E}(f(X_t)|X_0=x)</math>. In general, it is not easy to describe the domain of the Feller generator. However, the Feller generator is always closed and densely defined. If <math>X</math> is <math>\mathbb{R}^d</math>-valued and <math>D(A)</math> contains the test functions (compactly supported smooth functions) then[1] <math display="block">A f(x) = - c(x) f(x) + l (x) \cdot \nabla f(x) + \frac{1}{2} \text{div} Q(x) \nabla f(x) + \int_{\mathbb{R}^d \setminus{\{0\}}} \left( f(x+y)-f(x)-\nabla f(x) \cdot y \chi(|y|) \right) N(x,dy),</math> where <math>c(x) \geq 0</math>, and <math>(l(x), Q(x),N(x,\cdot))</math> is a Lévy triplet for fixed <math>x \in \mathbb{R}^d</math>.
Lévy processes
The generator of Lévy semigroup is of the form <math display="block">A f(x)= l \cdot \nabla f(x) + \frac{1}{2} \text{div} Q \nabla f(x) + \int_{\mathbb{R}^d \setminus{\{0\}}} \left( f(x+y)-f(x)-\nabla f(x) \cdot y \chi(|y|) \right) \nu(dy)</math> where <math>l \in \mathbb{R}^d, Q\in \mathbb{R}^{d\times d} </math> is positive semidefinite and <math>\nu </math> is a Lévy measure satisfying <math display="block">\int_{\mathbb{R}^d\setminus \{0\}} \min(|y|^2,1) \nu(dy) < \infty</math> and <math>0 \leq 1-\chi(s) \leq \kappa \min(s,1)</math>for some <math>\kappa >0</math> with <math>s \chi(s)</math> is bounded. If we define <math display="block">\psi(\xi)=\psi(0)-i l \cdot \xi + \frac{1}{2} \xi \cdot Q \xi + \int_{\mathbb{R}^d \setminus \{0\}} (1-e^{i y \cdot \xi}+i\xi \cdot y \chi(|y|)) \nu(dy )</math> for <math>\psi(0) \geq 0</math> then the generator can be written as <math display="block">A f (x) = - \int e^{i x \cdot \xi} \psi (\xi) \hat{f}(\xi) d \xi</math> where <math>\hat{f}</math> denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol <math>-\psi</math>.
Stochastic differential equations driven by Lévy processes
Let <math display="inline" id="L">L</math> be a Lévy process with symbol <math>\psi</math> (see above). Let <math>\Phi</math> be locally Lipschitz and bounded. The solution of the SDE <math>d X_t = \Phi(X_{t-}) d L_t</math> exists for each deterministic initial condition <math>x \in \mathbb{R}^d</math> and yields a Feller process with symbol <math>q(x,\xi)=\psi(\Phi^\top(x)\xi).</math>
Note that in general, the solution of an SDE driven by a Feller process which is not Lévy might fail to be Feller or even Markovian.
As a simple example consider <math display="inline">d X_t = l(X_t) dt+ \sigma(X_t) dW_t</math> with a Brownian motion driving noise. If we assume <math>l,\sigma</math> are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol <math display="block">q(x,\xi)=- i l(x)\cdot \xi + \frac{1}{2} \xi Q(x)\xi.</math>
Mean first passage time
The mean first passage time <math>T_1</math> satisfies <math>\mathcal A T_1 = -1</math>. This can be used to calculate, for example, the time it takes for a Brownian motion particle in a box to hit the boundary of the box, or the time it takes for a Brownian motion particle in a potential well to escape the well. Under certain assumptions, the escape time satisfies the Arrhenius equation.[2]
Generators of some common processes
For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix.
The general n-dimensional diffusion process <math>dX_t = \mu(X_t, t) \,dt + \sigma(X_t, t) \,dW_t</math> has generator<math display="block">\mathcal{A}f = (\nabla f)^T \mu + tr( (\nabla^2 f) D)</math>where <math>D = \frac 12 \sigma\sigma^T</math> is the diffusion matrix, <math>\nabla^2 f</math> is the Hessian of the function <math>f</math>, and <math>tr</math> is the matrix trace. Its adjoint operator is[2]<math display="block">\mathcal{A}^*f = -\sum_i \partial_i (f \mu_i) + \sum_{ij} \partial_{ij} (fD_{ij})</math>The following are commonly used special cases for the general n-dimensional diffusion process.
- Standard Brownian motion on <math>\mathbb{R}^{n}</math>, which satisfies the stochastic differential equation <math>dX_{t} = dB_{t}</math>, has generator <math display="inline">{1\over{2}}\Delta</math>, where <math>\Delta</math> denotes the Laplace operator.
- The two-dimensional process <math>Y</math> satisfying: <math display="block">\mathrm{d} Y_{t} = { \mathrm{d} t \choose \mathrm{d} B_{t} }</math> where <math>B</math> is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator: <math display="block">\mathcal{A}f(t, x) = \frac{\partial f}{\partial t} (t, x) + \frac1{2} \frac{\partial^{2} f}{\partial x^{2}} (t, x)</math>
- The Ornstein–Uhlenbeck process on <math>\mathbb{R}</math>, which satisfies the stochastic differential equation <math display="inline">dX_{t} = \theta(\mu-X_{t})dt + \sigma dB_{t}</math>, has generator: <math display="block">\mathcal{A} f(x) = \theta(\mu - x) f'(x) + \frac{\sigma^{2}}{2} f(x)</math>
- Similarly, the graph of the Ornstein–Uhlenbeck process has generator: <math display="block">\mathcal{A} f(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^{2} f}{\partial x^{2}} (t, x)</math>
- A geometric Brownian motion on <math>\mathbb{R}</math>, which satisfies the stochastic differential equation <math display="inline">dX_{t} = rX_{t}dt + \alpha X_{t}dB_{t}</math>, has generator: <math display="block">\mathcal{A} f(x) = r x f'(x) + \frac1{2} \alpha^{2} x^{2} f(x)</math>
See also
References
- Шаблон:Cite book (See Chapter 9)
- Шаблон:Cite book (See Section 7.3)
