How to compute (if it is possible) : $\mathbb{E}[\sup_{s \leq t} (\alpha + \beta e^{s} + \int_{0}^s e^{u} dB_u)^{-}]$ ? (where $(x)^- = \max(-x, 0)$ is the negative part and $\alpha, \beta \in \mathbb{R}^2$.
I suspect a Girsanov theorem to be somewhere, but I am really not familiar with this theorem. I wasn't able to apply it.
Thank you,
EDIT:
I think that I can handle the case, $\alpha = \beta = 0$. Indeed, in that case : $$\mathbb{E}[\sup_{s \leq t} (\int_0^t e^{u} dB_{u})^-] = \mathbb{E}[\sup_{s \leq t} (B_{\frac12(e^{2s} - 1)})^{-}] = \mathbb{E}[\sup_{s \leq t} B_{\frac12(e^{2s} - 1)}] = \frac1{\sqrt{\pi}} \sqrt{e^{2 t} -1}$$
EDIT :
In fact, the more general problem may help. I consider an Ornstein-Uhlenbeck process : $dY_t = (a + b Y_t) dt + \sigma dB_t$ where $a, b \in \mathbb{R}$ and $\sigma > 0$. And I would like to compute $\mathbb{E}[\sup_{s \leq t} (e^{a s} Y_s)^{-}]$. For the context, this appears in reflected Ornstein-Uhlenbeck process.