Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process $$ dr_t = ar_t dt + \sigma r_t dW_t, $$ does anyone know of an analytical bond price formula? We know that the time $t$ price of a pure discount $T$-bond $P(t,T)$ is $$ P(t,T) = E_Q\left(e^{-\int_t^T r_s \, ds}\right). $$ We also know that $$ r_t \mid r_0 \sim \log \mathcal{N}\left(\log r_0 + \left(a + \frac{\sigma^2}{2}\right)t, \sigma^2 t\right). $$ Now let $R := \int_t^T r_s ds$. The bond pricing equation becomes $$ P(t,T) = E_Q(e^{-R}), $$ so the questions is really, "what is the MGF for $-R$?" I'm having some trouble working this out for myself, in particular, what is the distribution of $R$?
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I edit this answer to give more details.
The process for $r$ above is the geometric Brownian motion (GBM) used to model stock prices in the Black-Scholes framework. Thus the question is about (the expectation of the) exponetial of the integral of GBM. The intergral of GBM is closely connected to Asian options. Thus one can study the literature about this topic.
According to this https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian3_doc.pdf
I would answer No: there is no closed formula for $P$ in your model.
