Search results for "Dynamic programming principle"
showing 8 items of 8 documents
Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities
2018
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in $\Omega\times \Omega$ via couplings.
Solutions of nonlinear PDEs in the sense of averages
2012
Abstract We characterize p-harmonic functions including p = 1 and p = ∞ by using mean value properties extending classical results of Privaloff from the linear case p = 2 to all pʼs. We describe a class of random tug-of-war games whose value functions approach p-harmonic functions as the step goes to zero for the full range 1 p ∞ .
Regularity for nonlinear stochastic games
2015
We establish regularity for functions satisfying a dynamic programming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations. peerReviewed
Hölder regularity for stochastic processes with bounded and measurable increments
2022
We obtain an asymptotic Hölder estimate for expectations of a quite general class of discrete stochastic processes. Such expectations can also be described as solutions to a dynamic programming principle or as solutions to discretized PDEs. The result, which is also generalized to functions satisfying Pucci-type inequalities for discrete extremal operators, is a counterpart to the Krylov-Safonov regularity result in PDEs. However, the discrete step size $\varepsilon$ has some crucial effects compared to the PDE setting. The proof combines analytic and probabilistic arguments.
Local regularity estimates for general discrete dynamic programming equations
2022
We obtain an analytic proof for asymptotic H\"older estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.
Asymptotic Hölder regularity for the ellipsoid process
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 next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.
Convergence of dynamic programming principles for the $p$-Laplacian
2018
We provide a unified strategy to show that solutions of dynamic programming principles associated to the $p$-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.
Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian
2021
AbstractWe prove a local Hölder estimate for any exponent $0<\delta <\frac {1}{2}$ 0 < δ < 1 2 for solutions of the dynamic programming principle $$ \begin{array}{@{}rcl@{}} u^{\varepsilon} (x) = \sum\limits_{j=1}^{n} \alpha_{j} \underset{\dim(S)=j}{\inf} \underset{|v|=1}{\underset{v\in S}{\sup}} \frac{u^{\varepsilon} (x + \varepsilon v) + u^{\varepsilon} (x - \varepsilon v)}{2} \end{array} $$ u ε ( x ) = ∑ j = 1 n α j inf dim ( S ) = j sup v ∈ S | v | = 1 u ε ( x + ε v ) + u ε ( x − ε v ) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $…