**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 adapted process $X$ is almost surely locally integrable, i.e. for any compact set $C \subset \mathbb R_+$we have $\int_C X_t \ dt < \infty$ a.s.

Define the moving average process $M$ by $M_t := \int_{[t, t + T]} X_s ds$.

Suppose that the following conditions hold:

Almost surely, $X_0 = X_s$ for all $s \leq T$.

$M_t$ is a $\mathcal H_t$-martingale.

Question:Is it true that $X$ is an $\mathcal F_t$-martingale?

notan $\mathcal H_t$-martingale. Indeed, $\mathbb E(M_{t+T}|\mathcal H_t) = \mathbb E(M_{t+T}|\mathcal F_{t+T}) = T X_{t+T}$ rather than $M_t$. $\endgroup$