FKG Inequality for Brownian Motion and Stochastic Differential Equations

Abstract

The purpose of this work is to study some possible application of FKG inequality to the Brownian motion and to Stochastic Differential Equations. We introduce a special ordering on the Wiener space and prove the FKG inequality with respect to this ordering. Then we apply this result on the solutions $X_t$ of a stochastic differential equation with a positive coefficient $\sigma$, we prove that these solutions $X_t$ are increasing with respect to the ordering, and finally we deduce a correlation inequality between the solution of different stochastic equations.