Exponential terminal moment below a linear Brownian boundary
= Exponential terminal moment below a linear Brownian boundary
{title2=$\mathbb E[e^{\theta W_t}\mathbf1_{\{T_{a,b}>t\}}]$}
For $T_{a,b}=\inf\{s:W_s=a+bs\}$ with $a>0$, exponential tilting of the terminal Brownian value gives $e^{\theta^2t/2}$ times the barrier-survival probability for Brownian drift $\theta-b$. Thus the moment equals $e^{\theta^2t/2}[\Phi((a+(b-\theta)t)/\sqrt t)-e^{2a(\theta-b)}\Phi((-a+(b-\theta)t)/\sqrt t)]$ for every real $\theta$.