$\widehat{S}^{-}(t)=\min_{t_0\leq t'\leq t} S(t')$, its minimum value, has not met the optimal boundary at or before $\tau$: $\widehat{S}^{\pm}(\tau)\lessgtr H_a $. In this case the problem of finding the passage time renews: $$\begin{aligned}
{\mathbb{E}}[e^{-r({t^{\pm}}_0-t_0)}{\mathbf 1}_{{t^{\pm}}_0> \tau}|{\mathcal{F}\/}(\tau)]= e^{-r(\tau-t_0)}{\mathbb{E}}[e^{-r({t^{\pm}}(\tau)-\tau)}|{\mathcal{F}\/}(\tau)]{\mathbf 1}_{\widehat{S}^{\pm}(\tau)\lessgtr H_a },
\label{Ftau}\end{aligned}$$ where the definition of ${t^{\pm}}(\tau)$ follows from (\[tpm\]).

The expected value in the right-hand side of (\[Ftau\]) is straightforward —cf (\[Eab\]): $$\begin{aligned}
{\mathbb{E}}[e^{-r({t^{\pm}}(\tau)-\tau)}|{\mathcal{F}\/}(\tau)]={\mathbb{E}_b}[e^{-r({t^{\pm}}(\tau)-\tau)}|{\mathcal{F}\/}(\tau)]=
\left(\frac{S(\tau)}{H_b}\right)^{\beta^{\pm}_b}.
$$ Therefore, we have to obtain the probability density function of $S(\tau)$, $S(\tau)={S_0}e^{R(\tau;t_0)}$, under the restriction $\widehat{S}^{\pm}(\tau)\lessgtr H_a$, in order to complete the computation of (\[EEEE\]). Note that, under the risk-neutral measure, we can express the return up to time $\tau$ as $R(\tau;t_0)=\pm \sigma_a \widetilde{W}^{\pm}_a(\tau;t_0)$, with $$\widetilde{W}^{\pm}_a(t;t_0)=\pm \left[W_a(t-t_0)+\frac{\theta_a}{\sigma_a}(t-t_0) \right],$$ a Wiener process with a drift. In these terms both conditions $\widehat{S}^{\pm}(\tau)\lessgtr H_a$ are equivalent to demand that the maximum of the Wiener process $\widetilde{M}^{\pm}_a(\tau;t_0)$ fulfils $$\widetilde{M}^{\pm}_a(\tau;t_0)=\max_{t_0\leq t'\leq \tau} \widetilde{W}^{\pm}_a(t';t_0) < \pm \frac{x_a}{\sigma_a}.$$ Thus, we have to compute the joint probability density function $\phi^{\pm}_a(m,w,t,t')$ of the two random processes, $$\begin{aligned}
{\mathbb{P}}\{ m<\widetilde{M}^{\pm}_a(t;t')\leq m+dm, w<\widetilde{W}^{\pm}_a(t;t')\leq w+dw\}\\
=\phi^{\pm}_a(m,w,t,t')dm dw,\end{aligned}$$ which reads: $$\begin{aligned}
\phi^{\pm}_a(m,w,t,t')=\nonumber \\
\frac{2(2m-w)}{\sqrt{2 \pi (t-t')^3}}\exp\left\{-\frac{(2m-w)^2}{2(t-t')} \pm\frac{\theta_a}{\sigma_a}w -\frac{\theta^2_a}{2 \sigma^2_a}(t-t')\right\}.\end{aligned}$$ Therefore $$\begin{aligned}
{\mathbb{E}}[e^{-r({t^{\pm}}-t_0)}{\mathbf 1}_{{t^{\pm}}_0> \tau}|{\mathcal{F}\/}(t_0)]=  
\frac{\lambda}{\lambda+\ell^{\pm}} \left[\left(\frac{{S_0}}{H_b}\right)^{\beta^{\pm}_b} -\left(\frac{H_a}{H_b}\right)^{\beta^{\pm}_b} \left(\frac{{S_0}}{H_a}\right)^{\gamma^{\pm}_{a}} \right],\end{aligned}$$ where we have introduced another constant depending on the 