Representation of Martingales with Jumps and Applications to Mathematical Finance

Abstract

We study representations of martingales with jumps based on the filtration generated by a Lévy process. Two types of representation theorem are obtained. The first formula is valid for any martingale and written as the sum of the stochastic integral based on the Brownian motion and that based on the compensated Poisson random measure. See (0.1). The second formula is valid only for a process which is a martingale for any equivalent martingale measure. See (0.2). The latter representation formula is then applied to a problem in mathematical finance. The upper hedging strategy and the lower hedging strategy of a contingent claim is obtained through the representation kernel.