Lecture 14: Ito calculus

Proof. Consider any sequence of partitions Πn = {0 = tn < tn < . . . < tn = 0 1 Nn T } such that Δ(Πn) = maxj |tn − tn| → 0. Additionally, suppose that the j+1 j sequence Πn is nested, in the sense the for every n1 ≤ n2, every point in Πn1 is also a point in Πn2 . Let X n = Xtn where j = max{i : ti ≤ t}. Then Xn is a t t j sub-martingale adopted to the same filtration (notice that this would not be the case if we instead chose right ends of the intervals). By the discrete version of the D-K inequality (see previous lectures), we have