Uniform Integrability of Nonnegative Supermartingales via Time Change in a Geometric Brownian Motion

Dedicated to Albert Nikolaevich Shiryaev, my teacher, on the occasion of his 90th birthday We propose a new approach to derivation of sufficient conditions for uniform integrability of nonnegative supermartingales or, equivalently, to derivation of conditions for absolute continuity of probability measures. Our approach is based on a representation of nonnegative supermartingales proposed by M. A. Urusov and the author of the present paper via a time change in a geometric Brownian motion. We prove a criterion for uniform integrability in terms of a time change and also put forward a new proof of sufficiency of the Novikov condition. It turns out that an elementary alternative proof of this fact can be reduced to its particular case for the stopped geometric Brownian motion.