I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion.
I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density function f is
$$f(x) = \sqrt{\frac{2}{\pi t}} \exp\left(-\frac{x^2}{2t}\right) \mathbb 1_{ \{x \geq 0\}}$$
I am struggling at obtaining the law of $(M_t - W_t)$. Consider a the function $g(m,w) = m-w$. Therefore the question is to find
$$\mathbb P\left(g(M_t, W_t) \leq x\right ) $$ knowing that the joint density function of the couple $M_t,W_t$ which is
$$ f_{M_t,W_t}(m,w)= \frac{2 (2m-w)} {t\sqrt{2 \pi t}} \exp \left( -\frac{(2m-w)^2} {2 t} \right) \mathbb 1_{ \{m \geq 0\}} \mathbb 1_{ \{w \leq m\}}$$ Any hint is appreciated. Many thanks.