 Home

Codex Wiki

Mathematics

Area of mathematics

Probability and statistics

Probability theory

Stochastic process

Brownian motion

Brownian reflection principle

Brownian running maximum

Let $M_t=\sup_{0\leq s\leq t}B_s$ for a standard <Brownian motion>. For $m\geq0$, the <Brownian reflection principle> gives
$$
\mathbb P(B_t\leq b,M_t\leq m)=
\begin{cases}
\Phi(b/\sqrt t)+\Phi((2m-b)/\sqrt t)-1,&b\leq m,\\
2\Phi(m/\sqrt t)-1,&b>m.
\end{cases}
$$
On $b<m$, the pair $(B_t,M_t)$ therefore has <joint probability density>
$$
f_{B_t,M_t}(b,m)=
\frac{2(2m-b)}{\sqrt{2\pi}\,t^{3/2}}
e^{-(2m-b)^2/(2t)}.
$$

Joint distribution of Brownian motion and its running maximum

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 201 / 5 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 29K / b / ii / Solution

Joint distribution of Brownian motion and its running maximum

 Ancestors (9)

 Incoming links (2)

 Discussion (0)

 Articles by others on the same topic (0)

Joint distribution of Brownian motion and its running maximum

 Ancestors (9)

 Incoming links (2)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)