Stochastic Integrals of Continuous Local Martingales, I
H. J. Engelbert, Juliaen Hess · Mathematische Nachrichten · 1980
Abstract Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǑ integral of X . Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n ‐dimensional case, and, in particular, to the n ‐dimensional time change of continuous local martingales with orthogonal components.