This book aims to widen the understanding of stochastic dynamic choice and equilibrium models. It offers a simplified and heuristic exposition of the theory of Brownian motion and its control or regulation, rendering such methods more accessible to economists who do not require a detailed, mathematical treatment of the subject. The main mathematical ideas are presented in a context which with which economists will be familiar. Using a binomial approach to Brownian motion, the mathematics is reduced to simple algebra, progressing to some equally simple limits. The starting point of the calculus of Brownian motion - 'Ito's Lemma' - emerges by analogy with the economics of risk-aversion. Conditions for the optimal regulation of Brownian motion, including the important, but often mysterious, 'smooth pasting' condition, are derived in a similar way. Each theoretical derivation is illustrated by developing a significant economic application, drawn mainly from recent research in macroeconomics and international economics.