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diffusion approximation; Markov chains; ito equation; difference equation; renewal process
The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable to the sequences of Markov chains $\left\{(^nX_m, ^nY_m), m=0,1,\ldots \right\}\ n=1,2,\ldots$ where the tendency of $\left\{^nY_m \right\}$ to the stationary state is greater than that of $\left\{^nX_m\right\}$.
[1] I. I. Gichman A. V. Skorochod: Theory of Random Processes III. (in Russian). Nauka, Moscow 1975. MR 0651014
[2] P. Mandl: A connection between Controlled Markov Chains and Martingales. Kybernetika 9 (1973), 237-241. MR 0323427 | Zbl 0265.60060
[3] P. Mandl: On aggregating controlled Markov chains. in Jaroslav Hájek Memorial Volume (to appear). MR 0561266 | Zbl 0431.93063
[4] P. Morton: On the Optimal Control of Stationary Diffusion Processes with Inaccessible Boundaries and no Discounting. J. Appl. Probability 8 (1971), 551 - 560. DOI 10.2307/3212178 | MR 0292572 | Zbl 0234.49019
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