A new kind of augmentation of filtrations

Abstract

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the $\sigma$-algebra generated by $(\mathcal{F}_t)_{t \geq 0}$) a coherent family of probability measures $(\mathbb{Q}_t)$, each of them being defined on $\mathcal{F}_t$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, but can have a negative answer if one takes its usual augmentation. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. In order to fix this problem, we introduce a new property for filtrations, intermediate between the right continuity and the usual conditions. We show that most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation, such as the existence of regular version of trajectories or the d\'ebut theorem, still hold under the N-augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration.