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The joint distribution of the running maximum

$ M_t = \max_{0 \leq s \leq t} W_s $

and $W_t$ is

$f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}}e^{-\frac{(2m-w)^2}{2t}}, m \ge 0, w \leq m $ (

Question is,

what is running maximum here? Is it same as If this is what "running maximum" is, I am not getting what $M_t$ exactly is referring to. Can anyone explain this?

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up vote 4 down vote accepted

Let $(W_t)_{t \geq 0}$ a Wiener Process on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. Then

$$M_t(\omega) := \max_{0 \leq s \leq t} W_s(\omega) \qquad (\omega \in \Omega)$$

is the running (pathwise!) maximum. This means that $M_t(\omega)$ is the maximum of the path $$[0,t] \ni s \mapsto W_s(\omega) \in \mathbb{R}$$

$\hspace{140pt}$Here is a picture...

The mapping $[0,\infty) \times \Omega \ni (t,\omega) \mapsto M(t,\omega)=M_t(\omega)$ is again a stochastic process.

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