The joint distributions of the brownian and both the minimum and the maximum respectively are known. What could be said about the joint distribution of the maximum and the minimum of a Brownian process?



The joint distribution of the min and max of a Brownian
I agree with @MattF. This seems like a standard question. Here is a reference by Biane, Pitmann and Yor
<https://arxiv.org/pdf/math/9912170.pdf>
Look at page 16, equation (56). It is expressed as a series similar to the Elliptic theta series that one gets for Brownian motion on a circle or using the reflection method for solutions of Laplace's equation on intervals.


Here is another perspective. Suppose that you allow that the distribution of $\tau\_{\{a,b\}}$ can be obtained.
Paraphrasing from McKean's Chapter 6.3.4 ``Two-sided passage times or Gambler's Ruin,'' in his textbook *Probability: The Classical Limit Theorems*,


<https://onlinelibrary.wiley.com/doi/abs/10.1111/insr.12297>


let $\tau\_{\{a,b\}}$ be the minimum of the passage times $\tau\_a \wedge \tau\_b$ to hit $a$ or $b$ assuming $-\infty<a<b<\infty$ and $B(0)=x$ with $x$ between $a$ and $b$.


Then an indirect way of getting the distribution of $\tau\_{\{a,b\}}$ is through its moment generating function, which is calculable with Doob's optional stopping theorem for a suitably constructed martingale:
$$
\mathbf{E}\_x\left[\exp(-\lambda \tau\_{\{a,b\}})\right]\,
=\, \frac{\cosh\left(\sqrt{2\lambda}\, (x-c)\right)}{\cosh\left(\sqrt{2\lambda}\, d\right)}\, ,
$$
$c$ being the midpoint $c=(a+b)/2$ and $d$ being the half-length $d=(b-a)/2$. See, for example, McKean's page 292.


Then
$\mathbf{P}\_x\big(\max(B([0,t])>a\, ,\ \min(B([0,t])>b\big)$ is the probability $B$ hits $\{b,a\}$ before time $t$, and it hits $a$ before $b$ (which is then $(x-b)/(a-b)$ by Gambler's ruin), and after the time$\tau\_{\{a,b\}}$ (at which $B$ hits $a$), it does not get down to $b$ before the extra time $t-\tau\_{\{a,b\}}$. This is the probability that an independent 1-sided passage time $\widetilde{\tau}\_{a-b}$ is greater than $t-\tau\_{\{a,b\}}$. If you had an explicit pdf for $\tau\_{\{a,b\}}$ that would reduce it to an integral, like a convolution for $\tau\_{\{a,b\}}$ and $\tau\_{\{a-b\}}$.


If you want a textbook that is freely available online, look at Chapter 2.4 of Morters and Peres, for example, [https://www.stat.berkeley.edu/~aldous/205B/bmbook.pdf](https://www.stat.berkeley.edu/%7Ealdous/205B/bmbook.pdf).


I do not have my copy of Varadhan's *Stochastic Processes* on hand. But it also seems like I remember that he also has an elliptic theta function in there for something, which might demonstrate the method to get formulas like the one that Biane, Pitman and Yor relate telegraphically.


