| In Part I of this paper, we isolate a class of Levy processes (with stationary, independent increments) X, such that every martingale with respect to the filtration generated by X can be represented as the sum of stochastic integrals with respect to the continuous martingale part and the purely discontinuous martingale part of X. In particular then, if X is a "mixture" of a Brownian motion and a compensated Poisson process, each such martingale can be written as a stochastic integral with respect to X itself, without any intervention of random measures. In Part II, we discuss the nonlinear filtering problem with Levy process observations, and extend results of Fujisaki-Kallianpur-Kunita (7) and Bremaud (2). The innovation process is introduced, and the martingale representation result is established for it; with the help of the latter, we establish the fundamental "filtering equations" that determine the evolution of the conditional distribution of the unobserved signal, given the past of the observations. In Part III, we apply the previous results to the study of hedging and optimization problems in financial economics models with jumps in the price system. The familiar "Equivalent Martingale Measure" is constructed, and leads to completeness of the financial market under a suitable non-degeneracy condition on the "noise matrix". Finally, utility maximization problems are introduced, and solved explicitly with the help of this theory; the results here generalize the work of Karatzas et. al. (14), (15) and that of Picque & Pontier (13). |
