Infinitesimal generator (stochastic processes)

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

General case ( according to Breiman )

Consider a time- homogeneous Markov process on a state space with transition semigroup, that is, for all is the corresponding transition kernels. Furthermore, it is the space of bounded, borel measurable functions, then each transition kernels can be interpreted as picture.

The infinitesimal generator of the process is the operator with domain

Which is given to all by

In detail this means that applies to all

With

Special case of a countable state space

Be a time- homogeneous Markov process with continuous time and discrete state space and transition semigroup with transition matrix for all.

Semigroup, intensity matrix Q matrix

The transition function or transition matrices form a semigroup because of the Chapman - Kolmogorov equations. [ They can be regarded as being above the space of bounded, measurable borel functions called illustrations. ]

Has the standard property and is called a standard transition function when

Or shortly

Does the default property, then for all: The figure is differentiable uniformly continuous and for all and, in Point 0 is the right -hand derivative

Written brief, we define this by

Called intensity matrix or simply matrix.

For all true, and applies to all with.

A state is called stable if, else instantly.

The transition function is called stable if all states are stable; in this case all entries of the associated intensity matrix are finite.

A state is called absorbing if, what exactly is the case if and only if for all.

The matrix Q and the corresponding Markov process are called conservative, if all row sums of Q are zero; this is exactly the case if and only if for all.

If Q is conservative, the process stable and the sequence diverges the jump times before reaching an absorbing state almost surely, the process is referred to as regular.

The entries can be interpreted as follows:

• Looking at the associated process to, you can specify with the help of the residence time in a state. This is - distributed, that is to apply. An absorbing state is then correspondingly an infinite residence time.
• It is true, the process is so "local poisson " and indicates small the rate at which the process of jumps in the state ( ).

About this interpretation it is often easier in practice, an appropriate Q matrix derive from the model assumptions, as directly specify, for example, M/M/1 / ∞ systems.

Uniformly continuous semigroup with infinitesimal generator

If the transition function is stable, it is a uniformly continuous semigroup whose infinitesimal generator Q. Can then be recovered from the behavior in infinitesimal time, the long-term behavior:

Said Matrixexponential called.

This is for example the case for a finite state space.

The stationary distribution of can then be a solution of the following system of equations

Determine which is interpreted as a row vector.

Generators of Feller processes

Feller processes are Markov processes in which the transition probabilities correspond qua a strongly continuous semigroup on the space of continuous, vanishing at infinity functions. In this case, the generator of the respective half- group

( defined for all of the limit with respect to the supremum norm exists ) is considered and the theorem of Hille - Yosida be applied.

Dynkins characteristic operator

The characteristic operator is a probabilistic representation of the analytic generator, which is often easier to work with. While forming in the above definition of the expected value of a fixed time (and subsequently goes to 0 ), in this case, the expected value of the various (random) time points is formed, to which the process a predetermined spatial area, for example a sphere around with radius leaves. For non-absorbing sets you

Absorbing sets for you. For large class of Feller processes applies maximum principle for continuous, vanishing at infinity functions due to Dynkins.

The definition and said connection goes back to a paper by EB Dynkin in 1955.