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Let $W_t$ be Brownian, and let $g$ be integrable , and odd, and subadditive $g(x+y)\le g(x)+g(y)$. How to show that $g(W_t)$ is a supermartingale? I am not sure how to make use of the subadditive of the function $g$. Can I have some hints please?

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Hint: For $0\leq s\leq t$, the subadditivity gives $g(W_t)\leq g(W_s)+g(W_t-W_s)$.

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