Transitional half group

In the theory of stochastic processes , the temporal change behavior of Markov processes is described by maps (with time parameters ) that form a so-called transition half-group , more precisely a half-group homomorphism . The change in the time interval can be broken down into the change during and the change during ( denote the sequential execution .) ${\displaystyle P^{t}}$ ${\displaystyle t\geq 0}$ ${\displaystyle [0,s+t]}$ ${\displaystyle [0,s]}$ ${\displaystyle [s,s+t].}$ ${\displaystyle \circ }$

${\displaystyle \forall \,s,t\in \mathbb {R} _{\geq 0}:\quad P^{[0,s]}\circ P^{[s,s+t]}=P^{[0,s+t]}}$.

In the case of processes that are homogeneous over time, the change is independent of and only depends on the length of the interval. The notation has the following property: ${\displaystyle P^{[s,s+t]}}$ ${\displaystyle s}$ ${\displaystyle t}$ ${\displaystyle P^{t}:=P^{[0,t]}(=P^{[s,s+t]})}$ ${\displaystyle (P_{t})}$

${\displaystyle \forall \,s,t\in \mathbb {R} _{\geq 0}:\quad P^{t}\circ P^{s}=P^{s+t}}$.

The composition of such images describing the change during time is therefore compatible with the addition of the time parameter. In other words, is a semigroup homomorphism between the semigroup formed by the time parameter and the addition operation and the semigroup ( transformation semigroup ). ${\displaystyle s}$ ${\displaystyle P^{s}}$ ${\displaystyle P}$ ${\displaystyle ([0,\infty ),+)}$ ${\displaystyle ((P^{s})_{s\geq 0},\circ )}$

In an abbreviated way of speaking, one speaks simply of a semigroup and describes the transition semigroup formed by the transition nuclei of a time-homogeneous Markov process. The compatibility of the addition in the time parameter and the sequential execution of nuclei is described by the Chapman-Kolmogorow equations . In this way, the definition of the transition semigroup makes it possible to apply the findings of semigroup theory to Markov processes.

Mathematical definition (in continuous time)

Be a temporally homogeneous Markov process in continuous time on a state space . Let the underlying probability space be and denote the expected value with respect to . ${\displaystyle (M_{t})_{t\geq 0}}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle (\Omega ,{\mathcal {A}},\mathbb {P} )}$ ${\displaystyle \mathbb {E} }$ ${\displaystyle \mathbb {P} }$

For all be and defined accordingly . ${\displaystyle x\in E}$ ${\displaystyle \mathbb {P} _{x}(\cdot ):=\mathbb {P} (\cdot \mid M_{0}=x)}$ ${\displaystyle \mathbb {E} _{x}(\cdot ):=\mathbb {E} (\cdot \mid M_{0}=x)}$

Be the transition cores . Then applies ${\displaystyle (P^{t})_{t\in \mathbb {R} _{\geq 0}}}$

${\displaystyle \forall \,t\in \mathbb {R} _{\geq 0}\ \forall \,x\in E\ \forall \,A\in {\mathcal {E}}:\quad P^{t}(x,A)=\mathbb {P} _{x}(M_{t}\in A)}$

With the Markov property the following Chapman-Kolmogorow equation then applies

${\displaystyle \forall \,s,t\in \mathbb {R} _{\geq 0}\ \forall \,x\in E\ \forall \,A\in {\mathcal {E}}:\quad P^{t+s}(x,A)=\mathbb {E} _{x}\mathbb {P} \left(M_{s+t}\in A\mid {\mathcal {F}}_{s}\right)=\mathbb {E} _{x}P^{t}(M_{s},A)=\int P^{t}(y,A)P^{s}(x,dy)\,,}$

which are briefly summarized in operator notation as

${\displaystyle \forall \,s,t\in \mathbb {R} _{\geq 0}:\quad P^{t+s}=P^{t}P^{s}.}$

They thus form a semi-group , which is referred to as a transition semi-group . Nothing is said about the topological properties of , therefore additional requirements are usually made on the Markov process, so that in a certain respect is continuous - for example in the case of the Feller processes , where a strongly continuous semigroup represents on . ${\displaystyle (P^{t})_{t\in \mathbb {R} _{\geq 0}}}$ ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle (P^{t})_{t\geq 0}}$ ${\displaystyle C_{0}}$

swell

• Sören Asmussen: Applied Probability and Queues. 2nd edition, Springer-Verlag, New-York 2003, ISBN 0387002111

Footnotes

1. ^ Asmussen, page 33