Conditional expectations (like, e.g., discounted prices in financial applications) are martingales under an appropriate filtration and probability measure. When the information flow arrives in a punctual way, a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates. Providing a way to find and construct piecewise constant martingales evolving in a connected subset of $\mathbb {R}$ is the purpose of this paper. After a brief review of possible standard techniques, we propose a construction scheme based on the sampling of latent martingales $\tilde {Z}$ with lazy clocks θ. These θ are time-change processes staying in arrears of the true time but that can synchronize at random times to the real (calendar) clock. This specific choice makes the resulting time-changed process $Z_{t}=\tilde {Z}_{\theta _{t}}$ a martingale (called a lazy martingale) without any assumption on $\tilde {Z}$ , and in most cases, the lazy clock θ is adapted to the filtration of the lazy martingale Z, so that sample paths of Z on [0,T] only requires sample paths of $\left (\theta, \tilde {Z}\right)$ up to T. This would not be the case if the stochastic clock θ could be ahead of the real clock, as is typically the case using standard time-change processes. The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on (interval of) $\mathbb {R}$ .

中文翻译：

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分段恒定mar和懒钟

有条件的期望（例如，金融应用程序中的折扣价格）是经过适当过滤和概率测度的mar。当信息流按时到达时，一个合理的假设是假设后者在信息更新的随机时间之间具有分段恒定的采样路径。提供寻找和构建在$ \ mathbb {R} $的连接子集中演化的分段恒定piece的方法是本文的目的。在简要回顾了可能的标准技术之后，我们提出了一种基于带有潜伏时钟θ的潜mar $ \ tilde {Z} $的采样的构造方案。这些θ是拖延真实时间的时变过程，但可以在随机时间与真实（日历）时钟同步。这种特定的选择使所得的时变过程$ Z_ {t} = \ tilde {Z} _ {\ theta _ {t}} $成为mar（称为惰性mar），而无需对$ \ tilde {Z} $进行任何假设，并且在大多数情况下，惰性时钟θ适用于惰性mar Z的过滤，因此Z在[0，T]上的采样路径仅需要$ \ left（\ theta，\ tilde {Z} \ right）$到T。如果随机时钟θ可以早于实际时钟，则情况并非如此，这通常是使用标准时变过程的情况。所提出的方法提供了一种简单的方法来构造在$ \ mathbb {R} $的（间隔）上演化的，易于分析的懒惰mar。惰性时钟θ适用于惰性mar Z的过滤，因此Z在[0，T]上的采样路径仅需要$ \ left（\ theta，\ tilde {Z} \ right）$到如果随机时钟θ可以早于实际时钟，则情况并非如此，这通常是使用标准时变过程的情况。所提出的方法提供了一种简单的方法来构造在$ \ mathbb {R} $的（间隔）上演化的可分析处理的惰性mar。惰性时钟θ适合于惰性mar Z的过滤，因此Z在[0，T]上的采样路径仅需要$ \ left（\ theta，\ tilde {Z} \ right）$到如果随机时钟θ可以早于实际时钟，则情况并非如此，这通常是使用标准时变过程的情况。所提出的方法提供了一种简单的方法来构造在$ \ mathbb {R} $的（间隔）上演化的，易于分析的懒惰mar。