Additive functionals of Markov processes and stochastic systems

Evgeny B. Dynkin

Annales de l'institut Fourier (1975)

  • Volume: 25, Issue: 3-4, page 177-200
  • ISSN: 0373-0956

Abstract

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Intuitively, an additive functional of a stochastic process ( x t , P ) gives a method to measure time taking into account the development of the process. We associate with any set of states C the mathematical expectation of time x t belongs to C . In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the δ -finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.

How to cite

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Dynkin, Evgeny B.. "Additive functionals of Markov processes and stochastic systems." Annales de l'institut Fourier 25.3-4 (1975): 177-200. <http://eudml.org/doc/74240>.

@article{Dynkin1975,
abstract = {Intuitively, an additive functional of a stochastic process $(x_t,P)$ gives a method to measure time taking into account the development of the process. We associate with any set of states $C$ the mathematical expectation of time $x_t$ belongs to $C$. In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the $\delta $-finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.},
author = {Dynkin, Evgeny B.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3-4},
pages = {177-200},
publisher = {Association des Annales de l'Institut Fourier},
title = {Additive functionals of Markov processes and stochastic systems},
url = {http://eudml.org/doc/74240},
volume = {25},
year = {1975},
}

TY - JOUR
AU - Dynkin, Evgeny B.
TI - Additive functionals of Markov processes and stochastic systems
JO - Annales de l'institut Fourier
PY - 1975
PB - Association des Annales de l'Institut Fourier
VL - 25
IS - 3-4
SP - 177
EP - 200
AB - Intuitively, an additive functional of a stochastic process $(x_t,P)$ gives a method to measure time taking into account the development of the process. We associate with any set of states $C$ the mathematical expectation of time $x_t$ belongs to $C$. In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the $\delta $-finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.
LA - eng
UR - http://eudml.org/doc/74240
ER -

References

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  1. [1]R. M. BLUMENTHAL and R. K. GETOOR, Markov Processes and Potential Theory, Academic Press, New York-London, 1968. Zbl0169.49204MR41 #9348
  2. [2]C. DELLACHERIE, Ensembles aléatoires I, Séminaire de Probabilités III, Université de Strasbourg, Lecture Notes in Math., 88, Springer-Verlag, Berlin-Heidelberg-New York, 1969, 97-114. Zbl0184.41202MR41 #2767
  3. [3]C. DELLACHERIE, Capacités et processus stochastiques, Springer-Verlag, Berlin-Heidelberg-New York, 1972. Zbl0246.60032MR56 #6810
  4. [4]E. B. DYNKIN, Markovskie processy, Fizmatgiz, Moskva, 1963, Markov Processes Vol. I-II, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. 
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  6. [6]E. B. DYNKIN, Additive functionals of Markov processes and their spectral measures, Dokl. Akad. Nauk SSSR, 214 (1974), 1241-1244 (English translation in Soviet Math. Dokl.). Zbl0306.60047MR49 #6376
  7. [7]E. B. DYNKIN, Markov representations of stochastic systems, Dokl. Akad. Nauk SSSR, 218 (1974), 1009-1012. Zbl0323.60070MR50 #14945
  8. [8]E. B. DYNKIN, Markov representations of stochastic systems, Uspehi Matem. Nauk, 30, 1 (1975), 61-99. Zbl0316.60019MR53 #6701
  9. [9]P. A. MEYER, Fonctionnelles multiplicatives et additives de Markov, Ann. Inst. Fourier, 12 (1962), 125-230. Zbl0138.40802MR25 #3570
  10. [10]P. A. MEYER, Processus de Markov, Lecture Notes in Math., 26, Springer-Verlag, Berlin-Heidelberg-New York, 1967. Zbl0189.51403MR36 #2219
  11. [11]P. A. MEYER, Probability and Potentials, Blaisdell, Waltham Mass., 1966. Zbl0138.10401MR34 #5119
  12. [12]M. METIVIER, Mesures vectorielles et intégrale stochastique II, Séminaire de probabilités, année 1972, Preprint. 
  13. [13]D. REVUZ, Mesures associées aux fonctionnelles additives de Markov I, Trans. Amer. Math. Soc., 148 (1970), 501-531. Zbl0266.60053MR43 #5611
  14. [14]M. ŠUR, Continuous additive functionals of Markov processes and excessive functions, Dokl, Akad. Nauk SSSR, 137 (1961), 800-803 [Soviet Math. 2 (1961), 365-368]. Zbl0111.33204MR26 #7023
  15. [15]M. ŠUR, On approximation of additive functionals of Markov processes Uspehi Mat. Nauk, 29, 6 (1974), 183-184. Zbl0326.60092
  16. [16]A. D. WENTZELL, Nonnegative additive functionals of Markov processes, Dokl. Akad. Nauk SSSR, 137 (1961), 17-20 [Soviet Math., 2 (1961), 218-221]. 

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