Gradient and Lipschitz Estimates for Tug-of-War Type Games
Attouchi, A., Luiro, H., & Parviainen, M. (2021). Gradient and Lipschitz Estimates for Tug-of-War Type Games. SIAM Journal on Mathematical Analysis, 53(2), 1295-1319. https://doi.org/10.1137/19M1256816
Published inSIAM Journal on Mathematical Analysis
© 2021, Society for Industrial and Applied Mathematics
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 gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in the higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as an improved version of the cylinder walk argument.
PublisherSociety for Industrial and Applied Mathematics
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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 ...
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)
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 ...
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