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On analytical methods for incomplete Markov random fields

Published online by Cambridge University Press:  01 July 2016

Timo Ravaska
Timo Ravaska*
Affiliation:
University of Oulu
*
∗Postal address: Department of Mathematics, Faculty of Science, University of Oulu, Linnanmaa, SF-90570 Oulu 57, Finland.
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Abstract

It is shown that many properties of Markov processes can be extended to the case where the time parameter is multidimensional (a partially ordered set). This includes Kolmogorov's backward and forward equations, and semi-group treatment of homogeneous processes.

Keywords

MARKOV PROPERTYMULTIDIMENSIONAL TIMETOTAL DIFFERENTIAL EQUATIONS (PFAFFIAN SYSTEMS)MULTIPARAMETER SEMI-GROUPSBIRTH-AND-DEATH PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 1 , March 1983 , pp. 99 - 112
Copyright
Copyright © Applied Probability Trust 1983 

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References

Blanc-Lapierre, A. and Fortet, R. (1953) Théorie des fonctions aléatoires. Masson, Paris.Google Scholar
Brockwell, P. J. (1965) Generalized one-dimensional scattering processes with applications in three dimensions. Nuovo Cimento 36, 370–384.CrossRefGoogle Scholar
Brockwell, P. J. and Moyal, J. E. (1964) Exact solutions of one-dimensional scattering problems. Nuovo Cimento 33, 776–796.Google Scholar
Cartan, H. (1971) Differential Forms. Hermann, Paris.Google Scholar
Cox, D. R. and Miller, H. D. (1965) Theory of Stochastic Processes. Methuen, London.Google Scholar
Cvintarnaja, N. D. (1980) Markov processes with a partially ordered set of time parameters. Theory Prob. Math. Statist. 21, 175–181.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
Dieudonne, J. (1960) Foundations of Modern Analysis. Academic Press, New York.Google Scholar
Feller, W. (1940) On the integro-differential equations of purely discontinuous Markoff processes. Trans. Amer. Math. Soc. 48, 488–515; 58, (1945) 474.Google Scholar
Feller, W. (1950) An Introduction to Probability Theory and Its Applications, Vol I. Wiley, New York.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York.Google Scholar
Hille, E. and Phillips, R. (1957) Functional Analysis and Semi-Groups. Amer. Math. Soc. Colloq. Publications XXXI, Providence, R1.Google Scholar
Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. University of Chicago Press.CrossRefGoogle Scholar
Kolmogorov, A. N. (1931) Ueber die analytischen Methoden in der Wahrscheinlichkeits-rechnung. Math. Ann. 104, 415–458.CrossRefGoogle Scholar
McCrea, W. H. and Whipple, F. J. W. (1940) Random paths in two and three dimensions. Proc. R. Soc. Edinburgh 60, 281–298.Google Scholar
Moyal, J. E. (1966) A general theory of first-passage distributions in transport and multiplicative processes. J. Math. Phys. 7, 464–473.Google Scholar
Nevanlinna, F. and Nevanlinna, R. (1973) Absolute Analysis. Springer-Verlag, Berlin.Google Scholar
Ravaska, T. (1981) On Markov processes with discrete states and multidimensional time. Report. Mathematics, University of Oulu, Finland.Google Scholar
Saaty, T. L. (1961) Some stochastic processes with absorbing barriers. J. R. Statist. Soc. B 23, 319–334.Google Scholar
Wendel, J. G. (1980) Hitting spheres with Brownian motion. Ann. Prob. 8, 164–169.Google Scholar