A Weak Convergence Criterion Constructing Changes of Measure
arXiv:1208.2606
Abstract
Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes.
15 pages, added a new example on sampling Ornstein-Uhlenbeck processes, conditioned on rare first passage time events