Brownian motion transform invariants

By Thijs van den Berg | Published: November 11, 2011

Below are some usefull transform invariants for Brownian motion.

    \begin{align*} \text{ $Y_t$ is also a }&\text{Brownian motion..}\\ \text{symmetry:}\\ Y_t&=-W_t\\ \\ \text{translation:}\\ Y_t&=W_t+a\\ \\ \text{scaling:}\\ Y_t&=c W_{t/c^2}\\ \\ \text{reflection principle:}\\ Y_t &= \left\{ \begin{tabular}{ll} $W_t$ & $t<T$ \\ $2W_t-W_t$ & $t>T$ \end{tabular} \right. \\ \\ \text{time inversion:}\\ Y_t &= \left\{ \begin{tabular}{ll} $0$ & $t=0$ \\ $t W_{1/t}$ & $t>0$ \end{tabular} \right. \\ \end{align*}

This entry was posted in Equations. Bookmark the permalink. Both comments and trackbacks are currently closed.