Geodesic
A controlled finite Markov chain with arbitrary set of decisions
Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 855-864 
R. Ya. Chitashvili. A controlled finite Markov chain with arbitrary set of decisions. Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 4, pp. 855-864. http://geodesic.mathdoc.fr/item/TVP_1975_20_4_a13/
@article{TVP_1975_20_4_a13,
     author = {R. Ya. Chitashvili},
     title = {A~controlled finite {Markov} chain with arbitrary set of decisions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {855--864},
     year = {1975},
     volume = {20},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1975_20_4_a13/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a controlled Markov chain with a finite set $S$ of states $s$ and an arbitrary set $A$ of decisions $a$ and with the optimality criterion of the form $$ \mathbf E^\pi\biggl[\sum_{n=1}^\tau r(s_n,a_n)+c(s_\tau,a_\tau)\biggr], $$ where the stopping moment $\tau$ does not depend on $(s_n,a_n);n\ge1)$ and has the geometric distribution. Sufficient conditions for the existence of $(k,\varepsilon)$-optimal policies are found.