FKG inequality for Brownian motion and stochastic differential equations. (English) Zbl 1060.60015

Summary: 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.


60E15 Inequalities; stochastic orderings
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65 Brownian motion
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