Generator (Markov processes)

The generator , generator , infinitesimal generator or infinitesimal generator of the transition semigroup of a time-homogeneous Markov process in continuous time is an operator which records the stochastic behavior of the process in infinitesimal time. Due to the Markov property and the temporal homogeneity, the process is determined or generated by its infinitesimal producer under certain conditions .

General case (after Breiman)

Given a time-homogeneous Markov process on a state space with a transition semigroup , that is , the corresponding transition kernel is for all . Furthermore, if the space of the limited, Borel measurable functions , then every transition kernel can be understood as a mapping . ${\displaystyle (M_{t})_{t\geq 0}}$ ${\displaystyle (E,{\mathfrak {E}})}$ ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle t\in \mathbb {R} _{\geq 0}}$ ${\displaystyle P^{t}}$ ${\displaystyle X}$ ${\displaystyle f\colon E\rightarrow \mathbb {R} }$ ${\displaystyle P^{t}\colon X\to X}$

The infinitesimal producer of the process is the operator with a domain ${\displaystyle A}$

${\displaystyle {\mathcal {D}}(A):=\left\{f\in X\;\left|\;\forall x\in E{\text{ existiert }}\lim _{t\downarrow 0}{\frac {P^{t}f(x)-f(x)}{t}}\right.\right\}}$,

which is given for all through ${\displaystyle f\in {\mathcal {D}}(A)}$

${\displaystyle Af=\lim _{t\downarrow 0}{\frac {P^{t}f-f}{t}}}$.

Detail, this means that for all true ${\displaystyle x\in E}$

${\displaystyle Af(x)=\lim _{t\downarrow 0}{\frac {P^{t}f(x)-f(x)}{t}}=\lim _{t\downarrow 0}{\frac {\operatorname {E} _{x}[f(M_{t})]-f(x)}{t}}}$

With

${\displaystyle P^{t}f(x)=\int f(y)P^{t}(x,dy)=\int f(y)\operatorname {P} _{x}^{M_{t}}(dy)=\operatorname {E} _{x}[f(M_{t})]}$.

The distribution of and denotes the expected value conditionally on the starting value . ${\displaystyle \operatorname {P} _{x}^{M_{t}}}$ ${\displaystyle M_{t}}$ ${\displaystyle \operatorname {E} _{x}}$ ${\displaystyle M_{0}=x\in E}$

Special case of a countable state space

Be a temporally homogeneous Markov process with continuous time, discrete state space and transition half-group with transition matrix for all . ${\displaystyle (M_{t})_{t\geq 0}}$ ${\displaystyle E}$ ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle P^{t}:=(p_{ij}(t))_{(i,j)\in E^{2}}}$ ${\displaystyle t\geq 0}$

Semigroup, intensity matrix, Q matrix

The transition function or transition matrices form a semigroup because of the Chapman-Kolmogorow equations . As above, they can be understood as mappings, denoting the space of the restricted, Borel measurable functions . ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle P^{t}\colon X\to X}$ ${\displaystyle X}$ ${\displaystyle f:E\rightarrow \mathbb {R} }$

${\displaystyle (P^{t})_{t\geq 0}}$ has the standard property or is called the standard transition function if

${\displaystyle \lim _{t\downarrow 0}p_{ij}(t)=p_{ij}(0)\;\;\;\forall (i,j)\in E^{2}}$

or short

${\displaystyle \lim _{t\downarrow 0}P^{t}=I}$

with the identity matrix . ${\displaystyle I}$

Has the standard property, the following applies to all : The mappings are uniformly continuous, differentiable for all and have the right-hand derivative at point 0 ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle (i,j)\in E^{2}}$
${\displaystyle t\mapsto p_{ij}(t)}$ ${\displaystyle t>0}$

${\displaystyle q_{ij}:=\lim _{t\downarrow 0}{\frac {p_{ij}(t)-p_{ij}(0)}{t}}\;\;\;\forall (i,j)\in E^{2}.}$

In short, this is defined by

${\displaystyle Q:=\lim _{t\downarrow 0}{\frac {P^{t}-I}{t}}.}$

${\displaystyle Q=(q_{ij})_{ij}}$is called the intensity matrix or simply Q-matrix .

For all true , and for all with valid . ${\displaystyle i\in E}$ ${\displaystyle q_{ii}\in [-\infty ,0]}$ ${\displaystyle i,j\in E}$ ${\displaystyle i\neq j}$ ${\displaystyle q_{ij}\in [0,\infty [}$

A state is called stable , if , otherwise instantaneously . ${\displaystyle i\in E}$ ${\displaystyle q_{ii}>-\infty }$

The transition function is said to be stable when all states are stable; in this case all entries of the associated intensity matrix are finite. ${\displaystyle (P^{t})_{t\geq 0}}$

A state is said to be absorbing if it applies, which is the case if and if it applies to all . ${\displaystyle i\in E}$ ${\displaystyle q_{ii}=0}$ ${\displaystyle p_{ii}(t)=1}$ ${\displaystyle t\geq 0}$

The matrix and its associated Markov process are said to be conservative if all row sums are zero; this is exactly the case if applies to all . ${\displaystyle Q}$ ${\displaystyle Q}$ ${\displaystyle \sum _{i\neq j}q_{ij}=-q_{ii}<\infty }$ ${\displaystyle i\in E}$

If the process is conservative, stable and the sequence of the jump times almost certainly diverges before an absorbing state is reached, the process is referred to as regular . ${\displaystyle Q}$

The entries can be interpreted as follows: ${\displaystyle q_{ij}}$

• If you consider the associated process, you can specify the dwell time in a state with the help of . This is exponentially distributed with an expected value , that is, applies to . An absorbing state then has an infinite dwell time. ${\displaystyle P_{t}}$ ${\displaystyle q_{ii}}$ ${\displaystyle i\in E}$ ${\displaystyle -{\tfrac {1}{q_{ii}}}}$ ${\displaystyle t,h>0,q_{ii}>-\infty }$ ${\displaystyle P(X_{s}=i,\forall s\colon t<s<t+h\mid X_{t}=i)=e^{hq_{ii}}}$
• It is true that the process is “locally poisson” and indicates the rate at which the process jumps from to the state ( ) for small . ${\displaystyle \quad p_{ij}(h)=q_{ij}h+o(h)}$ ${\displaystyle q_{ij}}$ ${\displaystyle h>0}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle i,j\in E,i\neq j}$

Using this interpretation, it is often easier in practice to derive a suitable Q-matrix from the model assumptions than to specify it directly, for example in the case of M / M / 1 / ∞ systems . ${\displaystyle P_{t}}$

Uniformly continuous semigroup with an infinitesimal generator

If the transition function is stable, it is a uniformly continuous semigroup whose infinitesimal generator is. Then the long-term behavior can be recovered from the behavior in infinitesimal time : ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle Q}$
${\displaystyle Q}$

${\displaystyle P(t)=e^{tQ}\quad {\text{für alle}}\quad t\geq 0}$,

where denotes the matrix exponential . This is the case, for example, for finite state spaces. The stationary distribution of can then be used as a solution to the following system of equations ${\displaystyle e}$ ${\displaystyle \pi }$ ${\displaystyle (P^{t})_{t\geq 0}}$

${\displaystyle \pi \cdot Q={\vec {0}}}$

determine, which is interpreted as a line vector. ${\displaystyle \pi }$

Generators from Feller processes

Feller processes are Markov processes in which the transition probabilities qua a strongly continuous semigroup on the space corresponding to the steady, the evanescent functions. In this case, the generator of the corresponding semigroup ${\displaystyle P^{t}(x,A)}$ ${\displaystyle (P^{t}f)(x):=\int P^{t}(x,dy)f(y)=\operatorname {E} _{x}f(M_{t})}$ ${\displaystyle C_{0}(E)}$

${\displaystyle Af={\underset {t\downarrow 0}{\operatorname {s-lim} }}{\frac {P^{t}f-f}{t}}}$

(defined for everyone for whom the limit value with regard to the supremacy norm exists) and the Hille-Yosida theorem applied. ${\displaystyle f\in C_{0}(E)}$

Dynkin's characteristic operator

The characteristic operator is a probabilistic equivalent of the analytical generator , which is often easier to work with. While in the above definition of the expected value of a fixed time is formed (and subsequently approaches 0), the expected value of is here (random) at the different time points is formed, to which the process a predetermined spatial area , for example a ball around with Radius , leaves. For what is not absorbing , one sets ${\displaystyle A}$ ${\displaystyle f(X_{t})}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle f(X_{\tau })}$ ${\displaystyle \tau =\tau (B)}$ ${\displaystyle B}$ ${\displaystyle B_{\nu ,x}}$ ${\displaystyle x=X_{0}}$ ${\displaystyle \nu }$ ${\displaystyle x}$

${\displaystyle (Uf)(x):=\lim _{\nu \to 0}{\frac {\operatorname {E} _{x}[f(X_{\tau (B_{\nu ,x})})]-f(x)}{\operatorname {E} _{x}[\tau (B_{\nu ,x})]}},}$

for absorbing one sets . For a large class of Feller processes we have continuous, infinitely vanishing functions due to Dynkin's maximum principle. ${\displaystyle x}$ ${\displaystyle (Uf)(x)=0}$ ${\displaystyle Af=Uf}$ ${\displaystyle f}$

The definition and the context mentioned go back to a work by EB Dynkin from 1955.

literature

• Leo Breiman: Probability. Addison-Wesley Publishing Company, Reading, Massachusetts 1968, ISBN 0-89871-296-3 .
• Bernt Øksendal: Stochastic Differential Equations. An Introduction with Applications. Springer Berlin 2003, ISBN 3-540-04758-1 .
• Daniel Revuz, Marc Yor: Continuous Martingales and Brownian Motion. Springer 2001, ISBN 3-540-64325-7 .
• Manuela Schmitz, Quasi-Stationarity in an Epidemiological Model, 2006 , Chapter 1.1 (PDF file; 418 kB).

Web links

Wiktionary: Generator  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ↑ Breiman, p. 377.
2. ↑ EB Dynkin: Infinitesimal operators of Markov stochastic processes , Doklady Akademii Nauk No. 105, 1955, pp. 206-209.