Rough path analysis : An Introduction

We denote by Bp,T (Rd) the Banach space which consists of continuous paths starting at 0 with finite p-variation norm ‖ ‖p. The 1-variation norm of x is the same as the total variation of x. The p-variation norm defines a weaker topology of the C1-path space with x0 = 0. Actually it is proved that I0,T (x) is a continuous functional of x in the p-variation topology for any 1 ≤ p < 2. However it is not a continuous function in general with respect to p-variation norm for p ≥ 2 if d ≥ 2. Terry Lyons [15] proved the following continuity results: Let 2 ≤ p < 3 and x and y be C1-path starting from 0. If the p-variation norm of x − y is small and the p/2-variation norm of the difference X2 and Y 2 (see the next section with respect to the definition of p/2-variation norm of them) is also small, then I0,T (x) − I0,T (y) is small. Here X2 s,t = ( X2,ij(s, t) ) 1≤i≤j≤d and