A localization of the L{\'e}vy operators arising in mathematical finances

TL;DR

This paper establishes the comparison principle and existence of viscosity solutions for integro-differential equations with Lévy operators, providing detailed Hölder continuity estimates in one dimension by localizing measure singularities.

Contribution

It introduces new methods for analyzing Lévy operators in integro-differential equations, including detailed regularity estimates and solution existence proofs.

Findings

Comparison principle established for Lévy operator equations

Existence of viscosity solutions demonstrated

Hölder continuity estimates obtained in one dimension

Abstract

The comparison principle and the existence of the solution of the integro-differential equation with L{\'e}vy operators, in the framework of the viscosity solution, are shown in this paper. For the one dimensional case, a detailed estimate of the H{\"o}lder continuity of solutions is presented, by localizing the singularity of the L{\'e}vy measure.