Let $M_t$ be a martingale on $\mathbb{Z}^d$ and assume $M_0=0$. Let $n \in \mathbb{N}$. For any $t \leq n$, let $Q_t^n$ be distributed as $|M_t|$ conditioned on $M_n=0$, where $|M_t|$ is some norm. Is $\{ Q_t^n \}_{1 \leq t \leq n}$ a supermartingale?
2025-01-13 00:04:55.1736726695
Is a martingale conditioned to return in the future a supermartingale?
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