A Maximal Inequality of the 2D Young Integral based on Bivariations

الأصل · EN

In this note, we establish a novel maximal inequality of the 2D Young integral ∫ₐᵇ∫cᵈ FdG in terms of the (p,q)-bivariation norms of the section functions x F(x,y) and y F(x,y) where G:[a,b]× [c,d]→ R is a controlled path satisfying finite (p,q)-variation conditions. The proof is reminiscent from the Young's original ideas young1 in defining two-parameter integrals in terms of (p,q)-finite bivariations. Our result complements the standard maximal inequality established by Towghi towghi1 in terms of joint variations. We apply the maximal inequality to get novel strong approximations for 2D Young integrals w.r.t the Brownian local time in terms of number of upcrossings of a given approximating random walk.