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discrete-time Markov control process; unbounded cost
For a discrete-time Markov control process with the transition probability $p$, we compare the total discounted costs $V_\beta $ $(\pi _\beta )$ and $V_\beta (\tilde{\pi }_\beta )$, when applying the optimal control policy $\pi _\beta $ and its approximation $\tilde{\pi }_\beta $. The policy $\tilde{\pi }_\beta $ is optimal for an approximating process with the transition probability $\tilde{p}$. A cost per stage for considered processes can be unbounded. Under certain ergodicity assumptions we establish the upper bound for the relative stability index $[V_\beta (\tilde{\pi }_\beta )-V_\beta (\pi _\beta )]/V_\beta (\pi _\beta )$. This bound does not depend on a discount factor $\beta \in (0,1)$ and this is given in terms of the total variation distance between $p$ and $\tilde{p}$.
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