Absolute continuity and singularity of locally absolutely continuous probability distributions.~I

Yu. M. Kabanov ; R. Sh. Liptser ; A. N. Shiryaev

Sbornik. Mathematics, Tome 35 (1979) no. 5, pp. 631-680

Résumé

Let $(\Omega,\mathscr F)$ be a measurable space provided with a nondecreasing family of $\sigma$-algebras ($\mathscr F_t)_{t\geqslant0}$ with $\mathscr F=\bigvee_{t\geqslant0}\mathscr F_t$ and $\widetilde{\mathsf P}$ and $\mathsf P$ two locally absolutely continuous probability measures on $(\Omega,\mathscr F)$, i.e., such that $\widetilde{\mathsf P}_t\ll\mathsf P_t$ for $t\geqslant0$ ($\widetilde{\mathsf P}_t$ and $\mathsf P_t$ are the restrictions of $\widetilde{\mathsf P}$ and $\mathsf P$ to $\mathscr F_t$). One asks when $\widetilde{\mathsf P}\ll \mathsf P$ or $\widetilde{\mathsf P}\perp\mathsf P$. An answer to this question is given in terms of the convergence set of a certain increasing predictable process constructed for the martingale $\mathfrak Z=(\mathfrak Z_t,\mathscr F_t,\mathsf P)$ with $\mathfrak Z_t=d\widetilde{\mathsf P}_t/d\mathsf P_t$. Actually, the somewhat more general situation of $\theta$-local absolute continuity of measures is studied. The proof of the fundamental theorem is based on a series of results that are of independent interest. In § 2 the theory of integration with respect to random measures is developed. § 4 deals with the convergence sets of semimartingales, and § 5 with the transformation of the predictable characteristics of a semimartingale under a locally absolutely continuous change of measure. Sufficient conditions are given in § 7 for the uniform integrability of nonnegative local martingales. Bibliography: 24 titles.