Some Classes of Discrete-Time Stochastic Processes

Abstract

The most important mathematical tools in pricing and hedging applications are certainly martingale and Markov properties. Martingality can be stated as a stochastic equation written in terms of conditional expectations. The Markov property can be stated in terms of deterministic semi-group Kolmogorov equations. The tower rule for conditional expectations and Markovian semi-group equations can be considered as primary dynamic programming pricing equations, in their stochastic and deterministic form (to mature as stochastic pricing BSDEs and deterministic pricing PIDEs later in the book). Practically speaking, a Markov property is a necessary companion to a martingale condition in order to ensure tractability (up to “the curse of dimensionality”). From an opposite perspective, martingales can be used to pursue some theoretical developments beyond a Markov setup. It’s interesting to note that historically, stochastic calculus was first developed as a tool for the study of Markov processes, until it was realized that the theory of martingales, in particular, was interesting for its own sake, allowing one to supersede Markovianity.

To emphasize these fundamental ideas at the simplest possible level of technicality, we present them in this first chapter in discrete time. We start with the definition and main properties of conditional expectation. The former is in fact mainly useful to prove the latter (which we leave to standard probability textbooks), since most practical conditional expectation computations are directly based on these properties, with the tower rule as a most emblematic example. We then introduce discrete time Markov chains and martingales in this spirit and with finance already in mind—basing most of our examples on random walks fortune processes, like with the doubling strategy, which grants a wealth of one for sure in the end but with values that are unbounded from below on the way!