Local gradient estimation for nonlinear degenerate parabolic variational inequalities derived from the value analysis of financial derivatives

This study examines the spatial gradient estimation of solutions to a particular class of variational inequality problems that arise from the structure of nonlinear degenerate parabolic operators. Specifically, under the assumption that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>2</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mn>2</m:mn> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mn>2</m:mn> <m:mfenced open="/" close=""> <m:mrow> <m:mi>n</m:mi> <m:mspace width="0.17em"/> </m:mrow> </m:mfenced> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> 2≤ p≤ 2(1+2/n .) (where n denotes the spatial dimension), we derive an estimate for the spatial gradient of the solution within a local cylindrical domain, referred to as the L p energy estimate, which is complemented by an additional L 2 estimate.