Editing Wiener process (section)

=== Running maximum ===
The joint distribution of the running maximum
<math display="block"> M_t = \max_{0 \leq s \leq t} W_s </math>
and {{math|''W<sub>t</sub>''}} is
<math display="block"> f_{M_t,W_t}(m,w) = \frac{2(2m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}}, \qquad m \ge 0, w \leq m.</math>

To get the unconditional distribution of <math>f_{M_t}</math>, integrate over {{math|−∞ < ''w'' ≤ ''m''}}:
<math display="block">\begin{align}
f_{M_t}(m) & = \int_{-\infty}^m f_{M_t,W_t}(m,w)\,dw = \int_{-\infty}^m \frac{2(2m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}} \,dw \\[5pt]
& = \sqrt{\frac{2}{\pi t}}e^{-\frac{m^2}{2t}}, \qquad m \ge 0,
\end{align}</math>

the probability density function of a [[Half-normal distribution]]. The expectation<ref>{{cite book|last=Shreve|first=Steven E| title=Stochastic Calculus for Finance II: Continuous Time Models|year=2008|publisher=Springer| isbn=978-0-387-40101-0| pages=114}}</ref> is
<math display="block"> \operatorname{E}[M_t] = \int_0^\infty m f_{M_t}(m)\,dm  = \int_0^\infty m \sqrt{\frac{2}{\pi t}}e^{-\frac{m^2}{2t}}\,dm = \sqrt{\frac{2t}{\pi}} </math>

If at time <math>t</math> the Wiener process has a known value <math>W_{t}</math>, it is possible to calculate the conditional probability distribution of the maximum in interval <math>[0, t]</math> (cf. [[Probability distribution of extreme points of a Wiener stochastic process]]). The  [[cumulative probability distribution function]] of the maximum value, [[Conditional probability|conditioned]] by the known value <math>W_t</math>, is:
<math display="block">\, F_{M_{W_t}} (m)  = \Pr \left( M_{W_t} = \max_{0 \leq s \leq t} W(s) \leq m \mid  W(t) = W_t \right) = \ 1 -\ e^{-2\frac{m(m - W_t)}{t}}\ \, , \,\ \   m > \max(0,W_t)</math>