Representation of Martingales, Quadratic Variation and Applications

In this paper, we present two related results. First, we shall obtain a sufficient condition under which a second order sample-continuous martingale can be represented as a stochastic integral in terms of a Brownian motion. Secondly, we shall show that if X and Y are sample-continuous local martingales (not necessarily with respect to the same family of (σ-algebras) and if either $X + Y$ or $X - Y$ is almost surely of bounded variation, then the quadratic variations of the two martingales are equal. This rather simple result has some surprising consequences.