Quadratic Variation. Consider the quadratic increments Next, for each n ≥ 1, let Vn(t) denote a similar quadratic increment sum for a partition 0 = tn0 n1 nkn = t, where maxk(tnk − tn,k−1) → 0. Show...

Quadratic Variation. Consider the quadratic increments
Next, for each n ≥ 1, let V_n(t) denote a similar quadratic increment sum for a partition 0 = t_n0
n1
<>_nk_n
= t, where maxk(tnk − tn,k−1) → 0. Show that E[Vn(t)2 − t] → 0 (which says Vn(t) converges in mean square distance to t). The function t is the quadratic variation of B in that it is the unique function (called a compensator) such that B(t) − t is a martingale. One can also show that V_n
→ t a.s. when the partitions are nested.