Chapman-kolmogorov Equation

$p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)$

$p\_\{i\_1,\backslash ldots,i\_\{n-1\}\}(f\_1,\backslash ldots,f\_\{n-1\})=\backslash int\_\{-\backslash infty\}^\{\backslash infty\}p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)\backslash ,df\_n$

Particularization to Markov chains

$p\_\{i\_1,\backslash ldots,i\_n\}(f\_1,\backslash ldots,f\_n)=p\_\{i\_1\}(f\_1)p\_\{i\_2;i\_1\}(f\_2\backslash mid\; f\_1)\backslash cdots\; p\_\{i\_n;i\_\{n-1\}\}(f\_n\backslash mid$

$p\_\{i\_3;i\_1\}(f\_3\backslash mid\; f\_1)=\backslash int\_\{-\backslash infty\}^\backslash infty\; p\_\{i\_3;i\_2\}(f\_3\backslash mid\; f\_2)p\_\{i\_2;i\_1\}(f\_2\backslash mid\; f\_1)df\_2.$

$P(t+s)=P(t)P(s)$

$P\_\{ij\}(t)=P(X\_t=j\backslash mid\; X\_0=i).$

See also

External links

• The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
• Mathworld

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