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diffusion approximation; Markov chains; ito equation; difference equation; renewal process

References:

[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