ABSTRACT

The statistical concept of causality in continuous time between filtered probability spaces considered in this paper is based on Granger's definition of causality. The given concept of causality can be connected to the purely discontinuous property for martingale and filtration. If M is a purely discontinuous local martingale with respect to purely discontinuous filtration $\left({\mathcal{G}}_t\right)$, we prove that M will remain purely discontinuous local martingale with respect to an extension $\left({\mathcal{F}}_t\right)$ of the filtration $\left({\mathcal{G}}_t\right)$ if and only if $\left({\mathcal{G}}_t\right)$ is its own cause within $\left({\mathcal{F}}_t\right)$. Moreover, we give conditions for a natural filtration of the purely discontinuous martingale $X_t$ to be a purely discontinuous filtration, using the property of causality. We also, consider the connection between the concept of statistical causality and the weak predictable representation property for purely discontinuous local martingales.

摘要

本文考虑的过滤概率空间之间连续时间因果关系的统计概念基于格兰杰对因果关系的定义。因果关系的给定概念可以与鞅和过滤的纯不连续性质联系起来。如果M是相对于纯不连续过滤的纯不连续局部鞅$\left({\mathcal{G}}_{吨}\right)$，我们证明M将保持纯粹不连续的局部鞅关于扩展$\left({\mathcal{F}}_{吨}\right)$ 过滤的 $\left({\mathcal{G}}_{吨}\right)$ 当且仅当 $\left({\mathcal{G}}_{吨}\right)$ 是它自己内在的原因 $\left({\mathcal{F}}_{吨}\right)$. 此外，我们给出了纯不连续鞅的自然过滤条件$X_{吨}$使用因果关系的属性是纯粹的不连续过滤。我们还考虑了统计因果关系的概念与纯不连续局部鞅的弱可预测表示特性之间的联系。