ABSTRACT
This chapter provides an introduction to continuous-time finance. It deals with the prototype of all continuous time models, and that is arithmetic Brownian motion (ABM). ABM is the most basic and important stochastic process in continuous time and continuous space, and it has many desirable properties including the strong Markov property, the martingale property, independent increments, normality, and continuous sample paths. From the perspective of modeling limited liability assets like common stocks, ABM can go negative, which does not make much economic sense. However, ABM is still the most basic stochastic process of its kind, so not too much should initially be made of the defect. In particular, it is possible to generate an option pricing formula for an underlying process that makes sense, from the option pricing formula for an underlying process that has some flaws.
