We give here a brief introduction to stochastic processes. Such processes do not appear explicitly in financial optimization models, however they are essential tools in asset pricing, and in the scenario generation methods that provide input data for the models. This chapter provides adequate coverage so that readers can follow the developments in the book when references are made to the use of stochastic processes. For additional material see the books by Bhat (1972), or Baxter and Rennie (2000).
A stochastic process is a family of random variables that are indexed by a parameter such as time or space. Modeling real-world phenomena as stochastic processes allows us to understand the behavior of the real-world situation. For example, we may want to study the behavior of a corporate security with respect to its credit rating. A corporate security may be in one of three possible states: it may have a high credit rating H, or be poorly rated P, or may go into default D. Analysis of firms' balance sheets will allow us to estimate the percentage of corporate securities that will remain in state H, or may be downgraded to P, or even go into default. The credit rating of the bond can be considered as a stochastic process which could be at one of the three states H, P, or D. Hence, this process is a discrete process. On the other hand, the number of corporate bonds that go into default is also a stochastic process, which takes a large number of possible ...