Extend Theorem 17.1 to local martingales with continuous paths.

Data From Theorem 17.1

17.1 Theorem (Doob-Meyer decomposition. Meyer 1962, 1963). For all square integ. Ex. 17.1 rable martingales Me M there exists a unique adapted, continuous and increasing process (M)=((M))st such that M - (M) is a martingale and (M)0 = 0. If (II)nal is any refining sequence of partitions of [0, T] with III 0, then (M) is given by (M), = lim (Ment, - Ment, 1), ty-1.jella where the limit is uniform (on compact t-sets) in probability. (17.1) The proof is split in a series of auxiliary results. This elementary approach is due to Kunita [152, Chapter 2.2]. We begin with the uniqueness of the decomposition. The following result will be useful in its own right.