Exponential terminal moment below a linear Brownian boundary ($\mathbb{E}[e^{\theta W_t}{1}_{\{T_{a,b}>t\}}]$)

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