= Maximum of a killed telegraph integral
{title2=$\mathbb P(\sup_{t\geq0}\int_0^tX_sds>c\mid X_0=1)=e^{-c\sqrt{\lambda(2q+\lambda)}}$}
Let $X$ switch between $\pm1$ at rate $q$ and be killed into zero at rate $\lambda$. To exceed a new level it must cross it in state $+1$. Restarting there by the <Strong Markov property> gives the multiplicative identity $\psi_+(c+d)=\psi_+(c)\psi_+(d)$, hence an exponential tail. First-step conditioning gives $\psi_-(c)=\int_0^\infty qe^{-(q+\lambda)u}\psi_+(c+u)du$ and $\psi_+(c)=e^{-(q+\lambda)c}+\int_0^cqe^{-(q+\lambda)u}\psi_-(c-u)du$. Substituting the exponential determines its rate as $\sqrt{\lambda(2q+\lambda)}$.
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