Abstract

The homogenization of the Lévy operator appears in various multi-scale models in finance, economy, biology, etc, which use jump-diffision process and pure jump process. The Lévy operator is the infinitesmal generator of such a process, in the following non-local form:

$$ Lu(x)= – int (R power N) [u(x+z) – u(x) – Du(x).z] d q(z), $$

where $q(.)$ the Lévy density represents the distribution of the length of jumps. When the density is symmetric, i.e. $q(z)=1/(|z| power (N+alpha))$ ($alpha in [0,2)$), the operator is known as the fractional power of the Laplacian. In applications, the Lévy densities are asymmetric in usual. We use the framework of the viscosity solution to treat such problems. In order to solve the homogenization of the integro-differential equation with the Lévy operator, we derive the ergodic cell peoblem. We prove the ergodicity of the jump-diffusion process and pure jump process in torus, by using the PDE method. The homegenization result is then proved rigorously.