While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question. Take a continuous-time stochastic process $X_t$ and define the the stopping time $T_a=\min\{t\geq0\,|\,X_t^{max}\geq a\}$, where $X_t^{max}=\max_{\tau\in[0,t]}(X_{\tau})$ and $a\in\mathbb{R}$. The well known result I was mentioning before is given by $\mathbb{P}(X_t^{max}\geq a)=\mathbb{P}(T_a\leq t)$ which leads to an Inverse Gaussian distribution with respect to time $t$.
Now take the afore-mentioned upper-bound limit $a$ and take a lower bound limit $b$; consider the stopping time $T_b=\min\{t\geq0\,|\,X_t^{min}\leq b\}$ with $b<a$, which is the first time the process goes below the limit $b$. Now the question is: what is the probability $\mathbb{P}(X_t^{max}\geq a \,\,\cap \,\, X_t^{min}\leq b)$? I think the procedure is, on the same line of the previous problem, to consider it in terms of stopping times, that is $\mathbb{P}(T_a\leq t \,\,\cap \,\, T_b\leq t)$, but do you have any idea on how to derive the distribution? To put it in words, I am looking for the probability that the process, in a time period $[0,t]$, both goes up $a$ and shrinks beneath $b$.
For a standard Brownian motion $B_t\,,$ (without drift), that starts in $x\in(0,a)\,,$ \begin{align} &\mathbb P_x\Big(B_s\in(0,a)\text{ for all }0\le s\le t\Big)\\\tag{1}&\quad=\sum_{k=-\infty}^\infty\textstyle\left\{\Phi\left(\frac{2ka+a-x}{\sqrt{t}}\right)-\Phi\left(\frac{2ka-x}{\sqrt{t}}\right)-\Phi\left(\frac{2ka+a+x}{\sqrt{t}}\right)+\Phi\left(\frac{2ka+x}{\sqrt{t}}\right)\right\} \end{align} (see Theorem 7.43 in this script). This probability can be formulated as $$ \mathbb P_x\Big(\min_{0\le s\le t} B_s>0\,\cap\,\max_{0\le s\le t} B_s<a\Big)\,. $$ For this BM the probability you are interested in is clearly \begin{align} &\mathbb P_x\Big(\min_{0\le s\le t} B_s\le 0\,\cap\,\max_{0\le s\le t} B_s\ge a\Big)=1-\mathbb P_x\Big(\min_{0\le s\le t} B_s>0\,\cup\,\max_{0\le s\le t} B_s<a\Big)\\ &=1-\mathbb P_x\Big(\min_{0\le s\le t} B_s>0\Big)-\mathbb P_x\Big(\max_{0\le s\le t} B_s<a\Big)+\underbrace{\mathbb P_x\Big(\min_{0\le s\le t} B_s>0\,\cap\,\max_{0\le s\le t} B_s<a\Big)}_{(1)}\,. \end{align} The other two terms in the last line are also known. The Girsanov theorem can be used to extend this to the Brownian motion with constant drift.