Let $B_t$ be a Brownian Motion. For $\mu>0$, $$\max_{0\le t\le T}(B_t+\mu t)\neq\max_{0\le t\le T}B_t+\mu T.$$ Why is this not equal in general? Is this because $B_t+\mu t$ is not a martingale?
2026-03-24 20:32:00.1774384320
Maximum of Brownian Motion with drift
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