On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian

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,wheretheorder$\frac 12$ 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.