Geodesic
Game-Type Random Walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 4, pp. 426-429

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We discuss a random walk in a convex set of Euclidean space ruled by two opponents. They may as usual independently choose a row and a column of the matrix of given random vectors. The surface of this set absorbs a moving point, and the payoff is defined in absorbation points. The determinateness of such games is proved with uniqueness theorems for Bellman-type functional equations under a somewhat artificial condition (cf. (66)). For the one-dimensional case (which is a generalization of Bellman–Milnor–Shapley’s “games of survival”) a more explicit analysis is given. Absorbation time is also considered as a payoff function both in one and multi-dimensional cases. 
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     author = {J. V. Romanovsky},
     title = {Game-Type {Random} {Walks}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {426--429},
     publisher = {mathdoc},
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     number = {4},
     year = {1961},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1961_6_4_a3/}
}
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J. V. Romanovsky. Game-Type Random Walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 4, pp. 426-429. http://geodesic.mathdoc.fr/item/TVP_1961_6_4_a3/