Stochastic Processes and Calculus

Abstract

By now there exist a number of books describing stochastic integrals and stochastic calculus in an accessible manner. Such introductory books, however, typically address an audience having previous knowledge about and interest in one of the following three fields exclusively: finance, econometrics, or mathematics. The textbook at hand attempts to provide an introduction into stochastic calculus and processes for students from each of these fields. Obviously, this can on no account be an exhaustive treatment. In the next chapter a survey of the topics covered is given. In particular, the book does neither deal with finance theory nor with statistical methods from the time series econometrician’s toolkit; it rather provides a mathematical background for those readers interested in these fields. The first part of this book is dedicated to discrete-time processes for modeling temporal dependence in time series. We begin with some basic principles of stochastics enabling us to define stochastic processes as families of random variables in general. We discuss models for short memory (so-called ARMA models), for long memory (fractional integration), and for conditional heteroscedasticity (socalled ARCH models) in respective chapters. One further chapter is concerned with the so-called frequency domain or spectral analysis that is often neglected in introductory books. Here, however, we propose an approach that is not technically too demanding. Throughout, we restrict ourselves to the consideration of stochastic properties and interpretation. The statistical issues of parameter estimation, testing, and model specification are not addressed due to space limitations; instead, we refer to, e.g., Mills and Markellos (2008), Kirchgässner, Wolters, and Hassler (2013), or Tsay (2005).