## Distribution built up from powers of a log normal r.v.

Let $W_t$ be a standard brownian motion, and $\lambda$ and $\alpha$ are positive constants. Does the expression $$\int_0^t \lambda e^{\lambda u + \frac{\alpha u X_t}{t} - \frac{\alpha^2 u^2}{2t}} du$$ have a recognizable distribution?

This comes from a quickest detection problem when a Brownian motion gains a draft at some unknown time. The expression below is part of the posterior probability that the drift has arrived, based on making a single observation at time $t$.

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you can do that integral somewhat explicitly by completing the square, and express is in terms of $\Phi$, the normal distribution function – mike May 28 at 23:35