1
Introduction

Stochastic differential equations (SDEs) are basically differential equations with an additional stochastic term. The deterministic term, which is common to ordinary differential equations, describes the 'average' dynamical behaviour of the phenomenon under study and the stochastic term describes the 'noise', i.e. the random perturbations that influence the phenomenon. Of course, in the particular case where such random perturbations are absent (deterministic case), the SDE becomes an ordinary differential equation.

As the dynamical behaviour of many natural phenomena can be described by differential equations, SDEs have important applications in basically all fields of science and technology whenever we need to consider random perturbations in the environmental conditions (environment taken here in a very broad sense) that affect such phenomena in a relevant manner.

As far as I know, the first SDE appeared in the literature in Uhlenbeck and Ornstein (1930). It is the Ornstein-Uhlenbeck model of Brownian motion, the solution of which is known as the Ornestein-Uhlenbeck process. Brownian motion is the irregular movement of particles suspended in a fluid, which was named after the botanist Brown, who first observed it at the microscope in the 19th century. The Ornstein-Ulhenbeck model improves Einstein treatment of Brownian motion. Einstein (1905) explained the phenomenon by the collisions of the particle with the molecules of the fluid and provided a model for the particle's position which corresponds to what was later called the Wiener process. The Wiener process and its relation with Brownian motion will be discussed on Chapters and .

Although the first SDE appeared in 1930, we had to wait till the mid of the 20th century to have a rigorous mathematical theory of SDEs by Itô (1951). Since then the theory has developed considerably and been applied to physics, astronomy, electronics, telecommunications, civil engineering, chemistry, seismology, oceanography, meteorology, biology, fisheries, economics, finance, etc. Using SDEs, one can study phenomena like the dispersion of a pollutant in water or in the air, the effect of noise on the transmission of telecommunication signals, the trajectory of an artificial satellite, the location of a ship, the thermal noise in an electric circuit, the dynamics of a chemical reaction, the control of an insulin delivery device, the dynamics of one or several populations of living beings when environmental random perturbations affect their growth rates, the optimization of fishing policies for fish populations subject to random environmental fluctuations, the variation of interest rates or of exchange rates, the behaviour of stock prices, the value of a call or put financial option or the risk immunization of investment portfolios or of pension plans, just to mention a few examples.

We will give special attention to the modelling issues, particularly the translation from the physical phenomenon to the SDE model and back. This will be illustrated with several examples, mainly in biological or financial applications. The dynamics of biological phenomena (particularly the dynamics of populations of living beings) and of financial phenomena, besides some clear trends, are frequently influenced by unpredictable components due to the complexity and variability of environmental or market conditions. Such phenomena are therefore particularly prone to benefit from the use of SDE models in their study and so we will prioritize examples of application in these fields. The study of population dynamics is also a field to which the author has dedicated a good deal of his research work. As for financial applications, it has been one of the most active research areas in the last decades, after the pioneering works of Black and Scholes (1973), Merton (1971), and Merton (1973). The 1997 Nobel prize in Economics was given to Merton and Scholes (Black had already died) for their work on what is now called financial mathematics, particularly for their work on the valuation of financial options based on the stochastic calculus this book will introduce you to. In both areas, there is a clear cross-fertilization between theory and applications, with the needs induced by applications having considerably contributed to the development of the theory.

This book is intended to be read by both more mathematically oriented readers and by readers from other areas of science with the usual knowledge of calculus, probability, and statistics, who can skip the more technical parts. Due to the introductory character of this presentation, we will introduce SDEs in the simplest possible context, avoiding clouding the important ideas which we want to convey with heavy technicalities or cumbersome notations, without compromising rigour and directing the reader to more specialized literature when appropriate. In particular, we will only study stochastic differential equations in which the perturbing noise is a continuous-time white noise. The use of white noise as a reasonable approximation of real perturbing noises has a great advantage: the cumulative noise (i.e. the integral of the noise) is the Wiener process, which has the nice and mathematically convenient property of having independent increments.

The Wiener process, rigorously studied by Wiener and Lévy after 1920 (some literature also calls it the Wiener-Lévy process), is also frequently named Brownian motion in the literature due to its association with the Einstein's first description of the Brownian motion of a particle suspended in a fluid in 1905. We personally prefer not to use this alternative naming since it identifies the physical phenomenon (the Brownian motion of particles) with its first mathematical model (the Wiener process), ignoring that there is an improved more realistic model (the Ornstein-Uhlenbeck process) of the same phenomenon. The 'invention' of the Wiener process is frequently attributed to Einstein, probably because it was thought he was the first one to use it (although at the time not yet under the name of 'Wiener process'). However, Bachelier (1900) had already used it as a (not very adequate) model for stock prices in the Paris Stock Market.

With the same concern of prioritizing simple contexts in order to more effectively convey the main ideas, we will deal first with unidimensional SDEs. But, of course, if one wishes to study several variables simultaneously (e.g. the value of several financial assets in the stock market or the size of several interacting populations), we need multidimensional SDEs (systems of SDEs). So, we will also present afterwards how to extend the study to the multidimensional case; with the exception of some special issues, the ideas are the same as in the unidimensional case with a slightly heavier matrix notation.

We assume the reader to be knowledgeable of basic probability and statistics as is common in many undergraduate degree studies. Of course, sometimes a few more advanced concepts in probability are required, as well as basic concepts in stochastic processes (random variables that change over time). Chapter intends to refresh the basic probabilistic concepts and present the more advanced concepts in probability that are required, as well as to provide a very brief introduction to basic concepts in stochastic processes. The readers already familiar with these issues may skip it. The other readers should obviously read it, focusing their attention on the main ideas and the intuitive meaning of the concepts, which we will convey without sacrificing rigour.

Throughout the remaining chapters of this book we will have the same concern of conveying the main ideas and intuitive meaning of concepts and results, and advise readers to focus on them. Of course, alongside this we will also present the technical definitions and theorems that translate such ideas and intuitions into a formal mathematical framework (which will be particularly useful for the more mathematically trained readers).

Chapter presents an example of an SDE that can be used to study the growth of a biological population in an environment with abundant resources and random perturbations that affect the population growth rate. The same model is known as the Black-Scholes model in the financial literature, where it is used to model the value of a stock in the stock market. This is a nice illustration of the universality of mathematics, but the reason for its presentation is to introduce the reader to the Wiener process and to SDEs in an informal manner.

Chapter studies the more relevant aspects of the Wiener process. Chapter?5 introduces the diffusion processes, which are in a certain way generalizations of the Wiener process and which are going to play a key role in the study of SDEs. Later, we will show that, under certain regularity conditions, diffusion processes and solutions of SDEs are equivalent.

Given an initial condition and an SDE, i.e. given a Cauchy problem, its solution is the solution of the associated stochastic integral equation. In a way, either in the deterministic case or in the case of a stochastic environment, a Cauchy problem is no more than an integral equation in disguise, since the integral equation is the fulcrum of the theoretical treatment. In the stochastic world, it is the integral version of the SDE that truly makes sense since derivatives, as we shall see, do not exist in the current sense (the...