Show that Proposition 17.2 and Corollary 17.3 remain valid for local martingales with continuous paths (cf. Definition 16.7).

Data From Proposition 17.2

Data From Corollary 17.3

Transcribed Image Text:

17.2 Proposition. Let (Xt, F)to be a martingale with paths which are continuous and of bounded variation on compact intervals. Then X = Xo a.s. for all t> 0. Proof. Without loss of generality we may assume that Xo = 0. Denote by V = VAR (X; [0,1]) and T, = inf{t >0 V, n} the strong variation of (X)20 and the stopping time when V, exceeds n for the first time. Since t X, is continuous, so ist V, cf. Paragraph A.29.f). https://doi.org/10.1515/9783110741278-017