Representation of martingales, quadratic variation and applications

Abstract. 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 a-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. 1. Introduction. Let {Xt,>-