Sobolev estimates and inverse Hölder estimates on a class of non-divergence variation-inequality problem arising in American option pricing

Abstract

We studied the Sobolev estimates and inverse Hölder estimates for a class of variational inequality problems involving divergence-type parabolic operator structures. These problems arise from the valuation analysis of American contingent claim problems. First, we analyzed the uniform continuity of the spatially averaged operator with respect to time in a spherical region and the Sobolev estimates for solutions of the variational inequality. Second, by using spatial and temporal truncation, we obtained the Caccioppoli estimate for the variational inequality and consequently derived the inverse Hölder estimate for the solutions.

Keywords

• variation-inequality problems
• non-divergence parabolic operator
• sobolev estimate
• inverse hölder estimate