Let $B_t$ a standard real valued Brownian motion, its running minimum $s_t \triangleq \min \limits_{0 \leq s \leq t} B_s$ and running maximum $S_t \triangleq \max \limits_{0 \leq s \leq t} B_s$. From Revuz Yor p.111, we know that for any Borel set $E \in \mathcal{B}([a,b])$, $$ \mathbb{P}(B_t \in E, a \leq s_t < S_t \leq b) = \int_E k(x) {\rm d} x $$ where $$ k(x) \triangleq \frac{1}{\sqrt{2 \pi t}}\sum \limits_{j \in \mathbb{Z}} f_j(x,a,b) $$ and for each $j$ $$ f_j(x,a,b) \triangleq \exp \left (-\frac{1}{2t}g_j(x,a,b)^2 \right ) - \exp \left (-\frac{1}{2t}\left (g_j(x,a,b) - 2b \right )^2 \right ) $$ and $g_j(x,a,b) \triangleq x+2j(b-a)$.
Question: Can we find a function $\hat k(x,\mathfrak{z}_1,\mathfrak{z}_2)$ such that $$ k(x) = \int_a^\infty \int_{\mathfrak{z}_1}^b \hat k(x,\mathfrak{z}_1,\mathfrak{z}_2) {\rm d} \mathfrak{z}_2 {\rm d} \mathfrak{z}_1? $$ In that case, the function $\hat k$ is the joint density of the triple $(B_t,s_t,S_t)$.
heads up on (my) difficulties
Maybe I am missing something but it does not seem straightforward. If we proceed term by term in the series, we would like to find $\hat f_j(x,\mathfrak{z}_1,\mathfrak{z}_2)$ such that $$ f_j(x,a,b) = \int_a^\infty \int_{\mathfrak{z}_1}^b \hat f_j(x,\mathfrak{z}_1,\mathfrak{z}_2) {\rm d} \mathfrak{z}_2 {\rm d} \mathfrak{z}_1, $$ thus $$ -\frac{\partial f_j}{\partial a}(x,a,b) = \int_a^b \hat f_j(x,a,\mathfrak{z}_2) {\rm d} \mathfrak{z}_2. $$ If we can find a function $\lambda_j(x,a,b)$ such that $$ -\frac{\partial f_j}{\partial a}(x,a,b) = \lambda_j(x,a,b) - \lambda_j(x,a,a), $$ then $\hat f_j$ must be $\frac{ \partial \lambda_j}{\partial b}$. If I am not mistaken, the problem is to find such a function $\lambda_j$.
From the formula $$ \mathbb{P}(B_t \in E, a \leq s_t < S_t \leq b) = \int_E k(x,a,b) {\rm d} x $$ where $$ k(x,a,b) \triangleq \frac{1}{\sqrt{2 \pi t}}\sum \limits_{j \in \mathbb{Z}} f_j(x,a,b) $$ you deduce directly that the joint density is $$\hat{k}(x,a,b)=-\frac{\partial^2 k}{\partial a \, \partial b} (x,a,b) $$ for $a<x<b$, as long as this partial derivative is continuous (which it is).