A variety of mathematical methods are applied to economists’ analyses of speculative pricing: general-equilibrium implicit equations akin to solutions for constrained-programming problems; difference equations perturbed by stochastic disturbances; the absolute Brownian motion of Bachelier of 1900, which anticipated and went beyond Einstein’s famous 1905 paper in deducing and analyzing the Fourier partial-differential equations of probability diffusion; the economic relative or geometric Brownian motion, in which the logarithms of ratios of successive prices are independently additive in the Wiener-Gauss manner, adduced to avoid the anomalies of Bachelier’s unlimited liability, and whose log-normal asymptotes lead to rational pricing functions for warrants and options which satisfy complicated boundary conditions; elucidation of the senses in which speculators’ anticipations cause price movements to be fair-game martingales; the theory of portfolio optimization in terms of maximizing expected total utility of all outcomes, in contrast to mean-variance approximations, and utilizing dynamic stochastic programming of Bellman-Pontryagin type; a molecular model of independent profit centers that rationalizes spontaneous buy-and-hold for the securities that exist to be held; a model of commodity pricing over time when harvests are a random variable, which does reproduce many observed patterns in futures markets and which leads to an ergodic probability distribution. Robert C. Merton provides a mathematical appendix on generalized Wiener processes in continuous time, making use of Ito formalisms and deducing Black-Scholes warrant-pricing functions dependent only on the certain interest rate and the common stock’s relative variance.