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Markov Processes and their Diffusion Approximations

  • Zeev Schuss
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 170)

Abstract

Recall that according to Definition 2.4.1, a stochastic process \(x(t)\) is a Markov process if for all times \(\tau_1\leq \tau_2 \leq \cdots \leq \tau_m \leq t_1 \leq t_2 \leq \cdots \leq t_n\) and all Borel sets \(A_1, A_2, \ldots , A_m, B_1, B_2,\ldots , B_n\) in \(\mathbb{R}^d\) its multidimensional conditionalPDF satisfies the equation
$$\begin{array}{ll}\ \ \ {\rm Pr}\{x(t_1) \in B_1, \ldots , x(t_n) \in B_n | x(\tau_1)\in A_1, \ldots , x(\tau_m) \in A_m\}\\ ={\rm Pr} \{x(t_1) \in B_1,\ldots , x(t_n) \in B_n | x(\tau_m) \in A_m\} .\end{array}$$
(7.1)

Keywords

Markov Process Master Equation Renewal Process Planck Equation Solvability Condition 
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 Science+Business Media, LLC 2010

Authors and Affiliations

  • Zeev Schuss
    • 1
  1. 1.Department of Applied MathematicsSchool of Mathematical Science Tel Aviv UniversityTel AvivIsrael

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