The Laplace transform of the first hitting time of Brownian motion
Use the fact that $0\leqslant M_t\mathbf{1}_{\{H_a>t\}}\leqslant\exp\left(\theta a−\frac12\theta^2t\right)$.
I m not sure that I understand properly the question.
We have $M_{H_a\wedge t}\to M_{H_a}$ almost surely, and
$M_{H_a\wedge t}<e^{\theta.a}$
The right hand side is constant so integrable, so doesn't the dominated convergence readily applies for $t \to +\infty$ ?
Am i missing something here ?
Regards