Regularity for nonlinear stochastic games

Abstract

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.2010 Mathematics Subject Classification. 91A15, 35J92, 35B65, 35J60, 49N60.

Asymptotic Hölder regularity for the ellipsoid process

“…In this section we compute the Taylor's expansion of f 1 in order to obtain the desired estimates for the comparison function. This is similar to [LP18,AHP17] and [ALPR]. However, for the convenience of the reader and expository reasons we write down the details.…”

“…The coupling approach to the regularity of different variants of tug-of-war with noise was first developed in [LP18] and applied in [PR16,AHP17,ALPR,Han]. As it turned out, for the continuous time diffusion processes and the Laplacian, the coupling method was utilized in connection to the regularity already at the beginning of 90's by Cranston [Cra91], utilizing the tools developed in [LR86].…”

Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

“…This argument plays an important role in obtaining the desired estimate. Several regularity results for functions satisfying various time-independent DPPs were proved by calculations based on this argument (see [LP18,AHP17,ALPR]). It was proved in [PR16] that functions satisfying another time-dependent DPP have Hölder regularity.…”

“…Our argument depends on the distance between two points. If two points are relatively far away, we will consider 'multidimensional DPP'(For a more detailed explanation, see [LP18]). We divide the argument into two subcases.…”

An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations

“…Indeed it is sufficient to reduce the problem to a ball of radius 1 for a non-constant function 0 < u < 1. The scheme of the proof, see for example, the idea in [17] or in [10] for a slightly different but equivalent approach, is the following one:…”