Ito-Wiener chaos expansion with exact residual and correlation, variance inequalities

Abstract

We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Ito-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdre-Kagan's variance inequality for Gaussian measure is also given.