Distribution of Trend Height and Length for Brownian Motion with a Drift

The concept of trend is one of the basic concepts in describing process behavior. There are at least several definitions of trends and often the notion of trend is used without making a precise definition. This work examines two definitions of trend, which are frequently used, and establishes the link between them. Moreover the description of distribution of trend height and length is given. Since Brownian motion is the benchmark model in financial mathematics these characteristics give the reference point with respect to which one should consider the empirical characteristics of trends.