A remark on the definitions of viscosity solutions for the integro-differential equations with L{\'e}vy operators

TL;DR

This paper demonstrates the equivalence of three definitions of viscosity solutions for integro-differential equations with Lévy operators, using an approximation technique to establish the result.

Contribution

It establishes the equivalence of multiple viscosity solution definitions for Lévy-involved equations, clarifying theoretical foundations.

Findings

Proves the equivalence of three viscosity solution definitions

Introduces a sequence of approximating test functions

Provides a key lemma for the approximation process

Abstract

The equivalence of three different definitions of viscosity solutions for the integro-differential equation with the L{\'e}vy operator is shown in this paper. The key is Lemma 2.1, in which we construct a sequence of the approximating test functions.