Itô’s Uniqueness and Uniqueness to the Martingale Problem

Abstract

The contents of this chapter are based on the work of Watanabe and Yamada [1971]. What we will be trying to do is give a careful comparison of the notion of uniqueness natural to the martingale problem as opposed to the notion of uniqueness inherent in Itô’s method. We have already seen indications that there is a distinction between these two notions (cf. Exercise 5.4.2) and that the distinction is intimately connected with questions of measurability of the path x(·) with respect to β(·) in the stochastic integral equation

$$ x(t) = x + \int\limits_s^{t \vee s} {\sigma (u,x(u))d\beta } (u) + \int\limits_s^{t \vee s} {b(u,x(u))du.} $$

(0.1)