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Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of $(\sup_{0\leq t\leq T}B_t^H,B_T^H)$, where $T$ is a fixed constant. I search this problem by google scholar, but only find some asymptotic results. No explicit results provided. Thanks for your attention.

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