On the local and global regularity of tug-of-war games
This thesis studies local and global regularity properties of a stochastic two-player zero-sum game called tug-of-war. In particular, we study value functions of the game locally as well as globally, that is, close to the boundaries of the game domains. Furthermore, we formulate a continuous time stochastic diﬀerential game and discuss, among other things, the equicontinuity of the families of value functions. The main motivation is to understand the properties of the games on their own right. As applications, we obtain an existence and a regularity result for a nonlinear elliptic p-Laplace type partial diﬀerential equation and a characterization of the solution to a parabolic p-Laplace type equation.
PublisherUniversity of Jyväskylä
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- Väitöskirjat 
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Heino, Joonas (International Statistical Institute; Bernoulli Society for Mathematical Statistics and Probability, 2018)We show that a uniform measure density condition implies game regularity for all 2 < p < ∞ in a stochastic game called “tug-of-war with noise”. The proof utilizes suitable choices of strategies combined with estimates for ...
Ruosteenoja, Eero (University of Jyväskylä, 2017)
Arroyo Garcia, Angel; Luiro, Hannes; Parviainen, Mikko; Ruosteenoja, Eero (Springer, 2020)We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ⊂ ℝn. The method of the proof is based on a game-theoretic idea to estimate the value of a related ...
Attouchi, Amal; Luiro, Hannes; Parviainen, Mikko (Society for Industrial and Applied Mathematics, 2021)We define a random step size tug-of-war game and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the ...
Arroyo, Ángel; Parviainen, Mikko (EDP Sciences, 2020)We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the ...