which may be shown to always exist. Note that in the special case of Brownian motion (or more generally a real-valued diffusion of the form d B = b ( t , B ) d t + d W where W is a Brownian motion), the term d [ B ] s simply reduces to d s , which explains why it is called the local time of B at x . For a discrete state-space process ( X s ) s ≥ ≥ 0 , the local time can be expressed more simply as