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• On the local and global regularity of tug-of-war games 

  Heino, Joonas (University of Jyväskylä, 2018)

  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 ...

• Uniform measure density condition and game regularity for tug-of-war games 

  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 ...

• Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities 

  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 ...

• Local regularity estimates for general discrete dynamic programming equations 

  Arroyo, Ángel; Blanc, Pablo; Parviainen, Mikko (Elsevier, 2022)

  We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for ...

• Gradient and Lipschitz Estimates for Tug-of-War Type Games 

  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 ...