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The $P$ dynamics of the underlying asset are: \begin{align*} dS=S(\mu dt+\sigma dB_t) \end{align*} That has the following solution under the $\mathcal{Q}$ dynamics: \begin{align*} S_t=S_0 e^{(r-\frac{\sigma^2}{2})t+\sigma W_t} \end{align*} Where $W_t$ is the equivalent martingale with respect to the original geometric brownian motion. Define $Y_t=\int_0^t ...


Yes it is known in closed form. See https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian_doc/asian_doc.html section 5.1 which references an older Geman-Yor paper.


A few tips. First note that $e^{-rt}S_t$ is a martingale. So make it appear and then integrate by part to rewrite $\int S_u du$ as a stochastic integral. Finally use the Ito isometry property.

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