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Arroyo, Ángel; Blanc, Pablo; Parviainen, Mikko

Hölder regularity for stochastic processes with bounded and measurable increments. (English) Zbl 1510.35087

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 40, No. 1, 215-258 (2023).
Summary: 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.

Cited in 2 Documents

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35J15 Second-order elliptic equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
91A50 Discrete-time games

Keywords:

dynamic programming principle; local Hölder estimates; stochastic process; equations in nondivergence form; \(p\)-harmonious; \(p\)-Laplace; tug-of-war games
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References:

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