As the title asks, does $|X_t|\leq K$ imply $\sup_{0\leq s\leq t}|X_t|\leq K$? Context: I am trying to prove that bounded continuous local martingales are martingales, and want to consider uniform integrability. I must admit that I can’t convince myself about the arguments for this simple problem!
2026-04-04 16:46:58.1775321218
Does $|X_t|\leq K$ imply $\sup_{0\leq s\leq t}|X_t|\leq K$?
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