Infinitesimal generator (stochastic processes) explained

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

\partial _{t} \rho = l A ^{*} \rho , where \rho is the probability density function, and l A ^{*} 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

(X _{t)} _{t} with Feller semigroup T = (T _{t)} _{t\geq} and state space E we define the generator (A,D(A)) by$$D(A)\; =\; \backslash left\backslash ,$$$$A\; f\; =\; \backslash lim\_\; \backslash frac,\; ~~\; \backslash text\; f\backslash in\; D(A).$$Here C _{0} (E) denotes the Banach space of continuous functions on E vanishing at infinity, equipped with the supremum norm, and T _{t} f(x) = E ^{x} f(X _{t)=E(f(X} _{t)|X} _{0=x)} . 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 X is R ^{d} -valued and D(A) contains the test functions (compactly supported smooth functions) then$$A\; f(x)\; =\; -\; c(x)\; f(x)\; +\; l\; (x)\; \backslash cdot\; \backslash nabla\; f(x)\; +\; \backslash frac\; \backslash text\; Q(x)\; \backslash nabla\; f(x)\; +\; \backslash int\_\; \backslash left(f(x+y)-f(x)-\backslash nabla\; f(x)\; \backslash cdot\; y\; \backslash chi(|y|)\; \backslash right)\; N(x,dy),$$where c(x) \geq 0 , and (l(x), Q(x),N(x, ⋅ )) is a Lévy triplet for fixed x \in R ^{d} .

Lévy processes

The generator of Lévy semigroup is of the form$$A\; f(x)=\; l\; \backslash cdot\; \backslash nabla\; f(x)\; +\; \backslash frac\; \backslash text\; Q\; \backslash nabla\; f(x)\; +\; \backslash int\_\; \backslash left(f(x+y)-f(x)-\backslash nabla\; f(x)\; \backslash cdot\; y\; \backslash chi(|y|)\; \backslash right)\; \backslash nu(dy)$$where

l \in R ^{d,} Q\in R ^{d x} is positive semidefinite and \nu is a Lévy measure satisfying and 0 \leq 1- \chi(s) \leq \kappa min (s, 1 ) for some \kappa > 0 with s \chi(s) is bounded. If we define$$\backslash psi(\backslash xi)=\backslash psi(0)-i\; l\; \backslash cdot\; \backslash xi\; +\; \backslash frac\; \backslash xi\; \backslash cdot\; Q\; \backslash xi\; +\; \backslash int\_\; (1-e^+i\backslash xi\; \backslash cdot\; y\; \backslash chi(|y|))\; \backslash nu(dy)$$for \psi( 0 ) \geq 0 then the generator can be written as$$A\; f\; (x)\; =\; -\; \backslash int\; e^\; \backslash psi\; (\backslash xi)\; \backslash hat(\backslash xi)\; d\; \backslash xi$$where \hat{f} denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol - \psi .

Stochastic differential equations driven by Lévy processes

Let $L$ be a Lévy process with symbol

\psi (see above). Let \Phi be locally Lipschitz and bounded. The solution of the SDE d X _{t} = \Phi(X _{t-} ) d L _{t} exists for each deterministic initial condition x \in R ^{d} and yields a Feller process with symbol q(x,\xi) = \psi(\Phi ^{\top(x)\xi).}

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 $d\; X\_t\; =\; l(X\_t)\; dt+\; \backslash sigma(X\_t)\; dW\_t$ with a Brownian motion driving noise. If we assume

l,\sigma are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol$$q(x,\backslash xi)=-\; i\; l(x)\backslash cdot\; \backslash xi\; +\; \backslash frac\; \backslash xi\; Q(x)\backslash xi.$$

Mean first passage time

The mean first passage time

T _{1} satisfies l A T _{1} = -1 . 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.

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

dX _{t} = \mu(X _{t,} t) dt + \sigma(X _{t,} t) dW _{t} has generator$$\backslash mathcalf\; =\; (\backslash nabla\; f)^T\; \backslash mu\; +\; tr((\backslash nabla^2\; f)\; D)$$where D = (12)/(\sigma\sigma) ^{T} is the diffusion matrix, \nabla ^{2} f is the Hessian of the function f , and tr is the matrix trace. Its adjoint operator is^[2] $$\backslash mathcal^*f\; =\; -\backslash sum\_i\; \backslash partial\_i\; (f\; \backslash mu\_i)\; +\; \backslash sum\_\; \backslash partial\_\; (fD\_)$$The following are commonly used special cases for the general n-dimensional diffusion process.

• Standard Brownian motion on

R ^{n} , which satisfies the stochastic differential equation dX _{t} = dB _{t} , has generator $\backslash Delta$, where \Delta denotes the Laplace operator.

• The two-dimensional process

Y satisfying: $$\backslash mathrm\; Y\_\; =$$ where B is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator: $$\backslash mathcalf(t,\; x)\; =\; \backslash frac\; (t,\; x)\; +\; \backslash frac1\; \backslash frac\; (t,\; x)$$

• The Ornstein–Uhlenbeck process on

R , which satisfies the stochastic differential equation $dX\_\; =\; \backslash theta(\backslash mu-X\_)dt\; +\; \backslash sigma\; dB\_$, has generator: $$\backslash mathcal\; f(x)\; =\; \backslash theta(\backslash mu\; -\; x)\; f\text{'}(x)\; +\; \backslash frac\; f$$(x)

• Similarly, the graph of the Ornstein–Uhlenbeck process has generator: $$\backslash mathcal\; f(t,\; x)\; =\; \backslash frac\; (t,\; x)\; +\; \backslash theta(\backslash mu\; -\; x)\; \backslash frac\; (t,\; x)\; +\; \backslash frac\; \backslash frac\; (t,\; x)$$
• A geometric Brownian motion on

R , which satisfies the stochastic differential equation $dX\_\; =\; rX\_dt\; +\; \backslash alpha\; X\_dB\_$, has generator: $$\backslash mathcal\; f(x)\; =\; r\; x\; f\text{'}(x)\; +\; \backslash frac1\; \backslash alpha^\; x^\; f$$(x)

See also

• Dynkin's formula

References

• Book: Calin, Ovidiu. An Informal Introduction to Stochastic Calculus with Applications. World Scientific Publishing. Singapore. 2015. 978-981-4678-93-3. 315. (See Chapter 9)
• Book: Øksendal , Bernt K. . Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications . Universitext. Sixth. Springer. Berlin . 2003 . 3-540-04758-1. 10.1007/978-3-642-14394-6. (See Section 7.3)

Notes and References

1. Book: Böttcher. Björn. Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Schilling. René. Wang. Jian. 2013. Springer International Publishing. 978-3-319-02683-1. en.
2. Web site: Lecture 10: Forward and Backward equations for SDEs . cims.nyu.edu.