Let $\{W_t\}_{t\geq 0}$ be a standard one-dimensional Brownian motion, and for each $t>0$ define $M_t=\max_{0\leq s \leq t} W_s$. Find the joint probability density of the random variables $(W_t, M_t)$.
Could anyone please help on how to proceed with this, I have solved his by assuming the two events are independent but i do't think that is the correct approach.
Best Answer
Using the reflection principle, the joint CDF of $(M_t, W_t)$ is $$ F_{t}(a,x)=\Phi\left(\frac{x}{\sqrt{t}}\right)+\Phi\left(\frac{2a-x}{\sqrt{t}}\right)-1, \quad a>0, a\ge x. $$
Differentiation $F_t$ w.r.t. $a$ and $x$ one gets $$ f_t(a,x)=\frac{2(2a-x)}{\sqrt{2\pi t^3}}e^{-\frac{(2a-x)^2}{2t}}. $$