Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Chapter 2Wiener Process

In mathematics, a Wiener process is a stochastic process sharing the same behaviour as Brownian motion, which is a physical phenomenon of random movement of particles suspended in a fluid. Generally, the terms “Brownian motion” and “Wiener process” are the same, although the former emphasises the physical aspects whilst the latter emphasises the mathematical aspects. In quantitative analysis, by drawing on the mathematical properties of Wiener processes to explain economic phenomena, financial information such as stock prices, commodity prices, interest rates, foreign exchange rates, etc. are treated as random quantities and then mathematical models are constructed to capture the randomness. Given these financial models are stochastic and continuous in nature, the Wiener process is usually employed to express the random component of the model. Before we discuss the models in depth, in this chapter we first look at the definition and basic properties of a Wiener process.

2.1 Introduction

By definition, a random walk is a mathematical formalisation of a trajectory that consists of taking successive random steps at every point in time. To construct a Wiener process in continuous time, we begin by setting up a symmetric random walk—such as tossing a fair coin infinitely many times where the probability of getting a head () or a tail () in each toss is . By ...