Overview
The representation theorem states that under a broad set of conditions, a martingale can be represented as an integral with respect to Brownian motion.
Theorem
For any proces {% f(t) \in \mathcal{L}^2 %} the following is a Martingale
{% M(t) = \int_0^t f(s) dW(s) %}
(see Bjork chpt 4)
Martingale Representation Theorem
Any Martingale adapted to a probability space generated by Brownian motions, can be represented as an integral over Brownian motion.
{% M(t) = M(0) + \int_0^t f(s) dW(s) %}