If $M_t$ is a continuous martingale, and $T$ arbitrary stopping time, is $M_t^T$ a martingale? Does this stay true when we only have right-continuity? I know it's true when the stopping time is bounded, but I am interested about this in the general case.
2026-04-03 13:32:53.1775223173
Is a stopped martingale always a martingale?
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Yes, if $(M_t)_{t\in\mathbb R_+}$ is a right-continuous martingale and $T$ is a stopping time, then $(M_{t\land T})_{t\in\mathbb R_+}$ also is a martingale, see for instance Corollary 3.24 in Le Gall, Brownian Motion, Martingales and Stochastic Calculus, 2016.