The Itô integral defined in Chapter 32 is a martingale with respect to the underlying filtration. The martingale representation theorem states that the converse is also true. In this chapter, we introduce the martingale representation theorem.

36.1 Basic Concepts and Facts

Theorem 36.1 (Martingale Representation Theorem). Let be the filtration generated by the Brownian motion , that is, be a square integrable martingale with respect to the filtration. Then Mt has a continuous version given by


where .

36.2 Problems

36.1. Let {Bt : t ≥ 0} be a Brownian motion with respect to a filtration . Let 0 ≤ a < b and X be a finite a-measurable random variable. ...

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