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The two systems of differential equations

  • Kai Lai Chung
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

Abstract

Let us recall the dual equations: if qi<∞,
$$ p'_{ij} \left( {s + t} \right) = \sum\limits_k {p'_{ik} } \left( s \right)p_{ij} \left( t \right); $$
(3.5 bis)
and if q i <∞,
$$ p'_{ij} \left( {t + s} \right) = \sum\limits_k {p_{ik} } \left( t \right)p'_{kj} \left( s \right); $$
(3.11 bis)
both valid for s > 0, t ≧ 0. The limiting cases for s = 0 may be written as
$$ p'_{ij} \left( t \right) = - q_i p_{ij} \left( t \right) + \sum\limits_{k \ne i} {q_{ik} } p_{kj} \left( t \right); $$
(1ij)
$$ p'_{ij} \left( t \right) = - p_{ij} (t)q_j + \sum\limits_{k \ne i} {p_{ik} } (t)q_{kj} ; $$
(2ij)

Keywords

Minimal Solution Continuous Parameter Sample Function Stationary Transition Probability Dual Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

Authors and Affiliations

  • Kai Lai Chung
    • 1
  1. 1.Syracuse UniversityUSA

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