Abstract
The theory of Markov processes is closely connected with the concept of Itô integrals for Wiener processes. In the two parameter case the fundamental definitions of Itô integrals and an Itô formula for two parameter stochastic processes were given by Ponomarenko (1) and Gichman (2). The Itô theory in the N parameter case was considered e.g. by Surgailis (3). An Itô calculus for N parameter, d dimensional stochastic processes was given by Imkeller (4). Using the basic concepts and notations of the two parameter Itô calculus, we derive Kolmogorov equations for several types of multiparameter Markov processes. We consider these equations for two parameter birth and death processes and two parameter diffusion processes. Two parameter semigroup properties are obtained. Applications are possible e.g. in the theory of reliability of systems with several components.
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5. References
Ponomarenko, L.L.: Stochastic integrals of several dimensional Brownian motion and related equations (in Russian); Probability theory and mathematical statistics 7(1972), 100–109
Gichman, Il.I.: On the Itô formula for two parameter stochastic integrals (in Russian); Theory of Random Processes 4(1976), 40–48
Surgailis,D.: On L2-and non-L2 multiple stochastic integration; in: Stochastic Differential Equations, Proc. 3rd IFIP-WG 7/1 Working Conf., Visegrad 1980, Springer-Verlag, Lecture Notes Control and Information 36, pp. 212–226
Imkeller, P.: Itô's formula for continuous (N,d)-processes; Z. Wahrscheinlichkeitstheorie verw. Gebiete 65(1984), 535–562
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© 1987 Springer-Verlag
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Hoy, L. (1987). Semigroup properties of markov processes with a several dimensional parameter. In: Engelbert, H.J., Schmidt, W. (eds) Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol 96. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038920
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DOI: https://doi.org/10.1007/BFb0038920
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