一种新的过滤强化方法
A new kind of augmentation of filtrations
摘要：
令$(Ω,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$是满足通常假设的过滤概率空间：通常不可能将概率测度$(\mathbb{Q}_t)$的相干族扩展到$\mathcal{F}_{\infty}$（由$(\mathcal{F}_t)_{t \geq 0}$生成的$σ$ -代数），其中每个概率测度定义在$\mathcal{F}_t$上。众所周知，例如，在维纳空间上，如果采用由坐标过程产生的过滤，则该扩展问题具有肯定的答案，但是如果采用其通常的增广，则可以具有否定的答案。另一方面，通常的假设是至关重要的，以获得最感兴趣的随机过程，如当地时间的标准布朗运动，随机积分等，为了解决这个问题，我们引入了一个新的属性过滤，中间的权利之间的连续性和通常条件的路径的存在。我们表明，大多数的随机过程理论的重要结果，一般证明下通常的增广，如存在定期版本的轨迹或首定理，仍然持有下的N-增广，而且这种新的增广允许扩展的连贯家庭的概率测度，只要这是可能的，与原来的过滤。
英文摘要：
Let $(Ω,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the $σ$-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 début 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.
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