Stochastic Integrals of Continuous Local Martingales, II

Abstract

AbstractConsider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n‐dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n‐dimensional time change for reducing every n‐dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.