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...y for cooperative games (i.e. the so called sup sup case). It is also worth mentioning that inspired by the “tug-of-war” (a discrete-time random turn game, see e.g. [37] and [31]), Atar and Budhiraja =-=[1]-=- studied a zero-sum stochastic differential game with U = V = {x ∈ Rn : |x| = 1} × [0,∞) played until the state process exits a given domain. The authors showed that the value of such a game is the un...

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...ns was established by Peres, Schramm, Sheffield and Wilson in [PSSW09], see also [LA98], [Obe05], and [LeG07]. This has inspired further studies to many different directions, see for example [PPS10], =-=[AB10]-=-, [MPRc] as well as led to simplified proofs in the theory of PDEs, see for example [AS10]. 2. Preliminaries Let us start by fixing the basic notation used throughout the work. We denote Bρ(x0) = {x ∈...

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...hn-Li [3]. It uses the theory of backward SDEs. An interesting connection between certain ”tug-of-war” stochastic differential games and the infinity Laplace equation was considered in Atar-Budhiraja =-=[1]-=- and Barron-Evans-Jensen [2]. 2 Stochastic differential games Given T > 0 a finite time horizon and t ∈ [0, T ), let (Ωt, F, Pt) be the canonical probability space, defined as Ωt = {ω ∈ C([t, T ]; R k...

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...nic functions verify an asymptotic mean value property, see, for example, [28], [13], [16], [21] and [22]. Concerning Tug-of-War games and PDEs the story begins with [33] and [34] and was extended in =-=[3]-=-, [4], [5], [31], etc. Remark that for the p−Laplacian it was proved in [14], [15] the equivalence between viscosity and weak solutions. This probability approach was used to obtain regularity propert...

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...Brownian motions. To understand the continuous time limit of the outlined price dynamics, as N → ∞, the key difficulty is to understand the asymptotic behavior of the term CNi (t). A solution, due to =-=[AB10]-=- in the context of time independent equations, is to replace √ N with dynamically controlled quantities d+ and d−. This approach is motivated by the connection between tug-of-war games and the infinit...

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... specified below. 1028 R. CAVAZOS-CADENA AND D. HERNÁNDEZ-HERNÁNDEZ The theory of games has interesting applications in diverse areas; see, for instance, Altman and Schwartz [1], Atar and Budhiraja =-=[2]-=-, and the recent book by Kolokoltsov and Malafeyev [5], and it should be mentioned that the topic of Markov Games was initiated in the pioneer papers by Shapley [12] and Zachrisson [17]. Also, stoppin...

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...D 2uDu−∆∞uDu), (10) is suggested in [26] as the game’s dynamics in the vanishing-ε limit. The relation is rigorously established in examples, but only heuristically justified in general. Recently, in =-=[3]-=-, a two-player zero-sum stochastic differential game (SDG) is considered, for which the value function uniquely solves (8) in the viscosity sense. The dynamics of the state process are given as Xt = x...

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...Du| 2 (D2u Du − ∆∞u Du), (1.3) is suggested in [7] as the game’s dynamics in the vanishing-ε limit. The relation is rigorously established in examples, but only heuristically justified in general. In =-=[3]-=-, a two-player zero-sum stochastic differential game (SDG) is considered, for which the value function uniquely solves (1.1) in the viscosity sense. The goal of the present paper is to show that, with...

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...| |Du| 2(D2uDu − ∆∞uDu), (1.3) is suggested in [7] as the game’s dynamics in the vanishing-ε limit. The relation is rigorously established in examples, but only heuristically justified in general. In =-=[3]-=-, a two-player zero-sum stochastic differential game (SDG) is considered, for which the value function uniquely solves (1.1) in the viscosity sense. The goal of the present paper is to show that, with...

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...oblem can be solved by a verification theorem instead of the dynamic programming principle. Inspired by the “tug-of-war” (a discrete-time random turn game, see e.g. [31] and [25]), Atar and Budhiraja =-=[1]-=- studied a zero-sum stochastic differential game with U = V = {x ∈ Rn : |x| = 1} × [0,∞) played until the state process exits a given domain. As in Chapter 6 of [22], the authors approximated such a g...