Chapman-Kolmogorov equation

The Chapman-Kolmogorow equation is in probability theory an equation for the transition probabilities in Markov chains or, more generally, in Markov processes . The differential notation of the Chapman-Kolmogorow equation is known as the master equation .

Markov chains

The Chapman-Kolmogorow equation for Markov chains represents the probability of the occurrence of the state after steps, starting in the state , as the sum of possible paths with an intermediate station . Formally, this means: ${\displaystyle y}$ ${\displaystyle m+n}$ ${\displaystyle x}$ ${\displaystyle z}$

Let be a Markov chain with transition matrix and state space . ${\displaystyle (X_{k})_{k\in \mathbb {N} _{0}}}$ ${\displaystyle \Pi }$ ${\displaystyle \mathrm {E} }$

Then applies to everyone ${\displaystyle x,y\in \mathrm {E} }$

${\displaystyle P(X_{m+n}=y\mid X_{0}=x)=\sum _{z\in \mathrm {E} }P(X_{m+n}=y\mid X_{m}=z)P(X_{m}=z\mid X_{0}=x)}$.

The proof of the equation is usually done as follows:

Applying the definition of matrix multiplication to the transition matrix results ${\displaystyle \Pi =(\Pi (x,y))_{x,y\in \mathrm {E} }}$

${\displaystyle {\begin{aligned}P(X_{m+n}=y\mid X_{0}=x)&{\overset {(*)}{=}}\Pi ^{n+m}(x,y)\\&{=}\sum _{z\in \mathrm {E} }\Pi ^{m}(x,z)\Pi ^{n}(z,y)\\&{\overset {(*)}{=}}\sum _{z\in \mathrm {E} }P(X_{m+n}=y\mid X_{m}=z)P(X_{m}=z\mid X_{0}=x)\,,\end{aligned}}}$

whereby it was exploited that applies to everyone with . ${\displaystyle (\ast )}$ ${\displaystyle P(X_{m+n}=y\mid X_{n}=x)=\Pi ^{m}(x,y)\,}$ ${\displaystyle m,n\in \mathbb {N} _{0},x,y\in \mathrm {E} \,}$ ${\displaystyle P(X_{n}=x)>0\,}$

Markov trials

For a general Markov process with the semigroup of transition nuclei, the Chapman-Kolmogorow equation can also be written briefly as ${\displaystyle (K(t))_{t\geq 0}}$

${\displaystyle \forall \,s,t\in \mathbb {R} _{\geq 0}:\quad K(s+t)=K(s)K(t)\,,}$

where denotes the composition of nuclei. Inductively it can be deduced from this that ${\displaystyle K(s)K(t)}$

${\displaystyle \forall \,n\in \mathbb {N} ,t_{1},\ldots ,t_{n}\in \mathbb {R} _{\geq 0}\quad K\left(\sum _{i=1}^{n}t_{i}\right)=\prod _{i=1}^{n}K(t_{i})\,.}$

Individual evidence

1. ↑ Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8 , p. 354.
2. ↑ Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8 , p. 291.