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Answer by user207651 for If the moving average of a process is a martingale,...
Let $X(t)$ be any deterministic $T$ periodic function, then$$M(t)=\int_0^T X(s) ds=\operatorname{Const},$$but $X(t)$ is not a martingale.
View ArticleIf the moving average of a process is a martingale, is the process a martingale?
Problem set up:Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.Suppose a cadlag...
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