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Harnack’s inequality for pharmonic functions via stochastic games
 Comm. Partial Differential Equations
, 1985
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On the value of stochastic differential games
, 2010
"... Abstract: We consider a two player, zero sum stochastic differential game based on a formulation given by Fleming and Souganidis. The saddle point property is introduced, and it is proved that the unique uniformly continuous bounded viscosity solution of the upper Isaacs PDE with boundary condition ..."
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Abstract: We consider a two player, zero sum stochastic differential game based on a formulation given by Fleming and Souganidis. The saddle point property is introduced, and it is proved that the unique uniformly continuous bounded viscosity solution of the upper Isaacs PDE with boundary condition satisfies such a property. Also, it is shown that approximately optimal Markov strategies can be constructed for both players. 1
MAXIMAL OPERATORS FOR THE pLAPLACIAN FAMILY
, 2015
"... We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appe ..."
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We prove existence and uniqueness of viscosity solutions for the following problem: max {−∆p1u(x), −∆p2u(x)} = f(x) in a bounded smooth domain Ω ⊂ RN with u = g on ∂Ω. Here −∆pu = (N + p)−1Du2−pdiv(Dup−2Du) is the 1homogeneous p−Laplacian and we assume that 2 ≤ p1, p2 ≤ ∞. This equation appears naturally when one considers a tugofwar game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual TugofWar game (without noise) or to play at random. Moreover, the operator max {−∆p1u(x), −∆p2u(x)} provides a natural analogous with respect to p−Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.
NASH EQUILIBRIA IN A CLASS OF MARKOV STOPPING GAMES
"... This work concerns a class of discretetime, zerosum games with two players and Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I and, if the system is no halted, player I selects an action to drive the system and rec ..."
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This work concerns a class of discretetime, zerosum games with two players and Markov transitions on a denumerable space. At each decision time player II can stop the system paying a terminal reward to player I and, if the system is no halted, player I selects an action to drive the system and receives a running reward from player II. Measuring the performance of a pair of decision strategies by the total expected discounted reward, under standard continuitycompactness conditions it is shown that this stopping game has a value function which is characterized by an equilibrium equation, and such a result is used to establish the existence of a Nash equilibrium. Also, the method of successive approximations is used to construct approximate Nash equilibria for the game.
Critical Branching Processes, (ii)Large Deviation Properties of Weakly Interacting Processes, (iii)Multiscale Diffusion Approximations for Stochastic Networks in Heavy Traffic, (iv) Adaptive Ergodic Control of Markov Chains, (v)Controlled Stochastic Netwo
, 2010
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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggesstions for reducing this burden, to Washington
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, 2008
"... On near optimal trajectories for a game associated with the ∞Laplacian ..."
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