Elements of the Theory of Markov Processes and Their ApplicationsThis graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition. |
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Page 63
... infinitesimal transition probabilities of the Markov process { X ( t ) , t > 0 } .1 If we now put P ( t ) = ( P ,, ( t ) ) , the Kolmogorov equations can be written in the form dP ( t ) dt dP ( t ) -- P ( t ) A ( t ) A ( t ) P ( t ) ...
... infinitesimal transition probabilities of the Markov process { X ( t ) , t > 0 } .1 If we now put P ( t ) = ( P ,, ( t ) ) , the Kolmogorov equations can be written in the form dP ( t ) dt dP ( t ) -- P ( t ) A ( t ) A ( t ) P ( t ) ...
Page 76
... infinitesimal transition probabilities is given by that is , -λ λ 0000 0 -λ 2000 0 0 2 2 0 0 Α αij = - 0 0 ... 0 0 0 0 оо одоо ... 0 0 λ for j otherwise *** i + 1 , i 0 , 1 , . . . ( 2.97 ) a i i -λ From the above we can immediately ...
... infinitesimal transition probabilities is given by that is , -λ λ 0000 0 -λ 2000 0 0 2 2 0 0 Α αij = - 0 0 ... 0 0 0 0 оо одоо ... 0 0 λ for j otherwise *** i + 1 , i 0 , 1 , . . . ( 2.97 ) a i i -λ From the above we can immediately ...
Page 79
... birth process , we have Li λί = for i = 1 , 2 , ... Qis - 1 for j = i + 1 , i = 1 , 2 , . . . = = 0 otherwise ( 2.114 ) Therefore , the matrix of infinitesimal transition probabilities is given CONTINUOUS TIME PROCESSES 79.
... birth process , we have Li λί = for i = 1 , 2 , ... Qis - 1 for j = i + 1 , i = 1 , 2 , . . . = = 0 otherwise ( 2.114 ) Therefore , the matrix of infinitesimal transition probabilities is given CONTINUOUS TIME PROCESSES 79.
Page 80
A. T. Bharucha-Reid. Therefore , the matrix of infinitesimal transition probabilities is given by 0 0 0 0 0 0 ... 0 -2 λ 0 0 0 ... A 0 0 -21 21 0 0 0 0 0 -32 32 0 Hence the Kolmogorov equations for the birth process are given by dPij ( t ) ...
A. T. Bharucha-Reid. Therefore , the matrix of infinitesimal transition probabilities is given by 0 0 0 0 0 0 ... 0 -2 λ 0 0 0 ... A 0 0 -21 21 0 0 0 0 0 -32 32 0 Hence the Kolmogorov equations for the birth process are given by dPij ( t ) ...
Page 84
... infinitesimal transition probabilities is also a function of time ; thus 1 1 -λ λ 0 0 ... 0 1 + aλt 1 + aλt 1 + α 1+ α 0 --λ λ 0 0 - A A ( t ) 1 + aλt 1 + αλι 1 + 2α 1 + 2α 0 0 λ 1 + αλί 1 + aλt Hence the Kolmogorov equations for the ...
... infinitesimal transition probabilities is also a function of time ; thus 1 1 -λ λ 0 0 ... 0 1 + aλt 1 + aλt 1 + α 1+ α 0 --λ λ 0 0 - A A ( t ) 1 + aλt 1 + αλι 1 + 2α 1 + 2α 0 0 λ 1 + αλί 1 + aλt Hence the Kolmogorov equations for the ...
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Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |
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