On the differentiability of the solutions of non-local Isaacs equations involving $\frac 12$-Laplacian

TL;DR

This paper establishes $C^{1,\sigma}$ regularity for solutions of non-local Isaacs equations involving the critical $rac{1}{2}$-Laplacian, enabling classical solvability in stochastic differential games with jump-diffusions.

Contribution

It proves the differentiability of viscosity solutions for non-local Isaacs equations with the critical fractional Laplacian, a novel result for non-translation invariant equations.

Findings

Solutions are $C^{1,\sigma}$ regular.

Viscosity solutions are classically solvable.

Addresses equations with critical fractional order $rac{1}{2}$-Laplacian.

Abstract

We derive $C^{1,\sigma}$-estimate for the solutions of a class of non-local elliptic Bellman-Isaacs equations. These equations are fully nonlinear and are associated with infinite horizon stochastic differential game problems involving jump-diffusions. The non-locality is represented by the presence of fractional order diffusion term and we deal with the particular case of $\frac 12$-Laplacian, where the order $\frac 12$ is known as the critical order in this context. More importantly, these equations are not translation invariant and we prove that the viscosity solutions of such equations are $C^{1,\sigma}$, making the equations classically solvable.