Sobolev estimates and inverse Ho<spacing diaeresis>lder estimates on a class of non-divergence variation-inequality problem arising in American option pricing

We studied the Sobolev estimates and inverse Ho<spacing diaeresis>lder 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 Ho<spacing diaeresis>lder estimate for the solutions.