# Basic Stochastic Processes : Review

This book is one of the few books on stochastic processes that teach concepts via problems. Teaching math concepts via problems with out delving too much in to the theory has its own advantages. The fact that the book is asking you to prove/compute/calculate/verify something at regular intervals means that you are not a passive reader from the word “go”. In fact one cannot be a passive reader at least while going over a math book. But this approach of “using problems” to teach various concepts is extremely appealing for people who are looking for self-study texts. There is a whole section on Markov chains where a few definitions theorems are interlaced between what is largely a set of interesting exercises that guide the reader to understand discrete and continuous Markov processes. The interesting thing about this book is that, even though the title seems to communicate that it is a “learn by doing” book, the math for all the proofs, lemma and propositions are quite rigorous. There are a few places where the author refers the reader to other books, but by and large it is a self-contained text.

The book starts with a basic recap of probability theory and covers sigma algebra, measures, probability space, borel measurable functions, sigma generated by a random variable etc. Well, the chapter is more a formality. If you are not comfortable with measure theory, this shouldn’t be your first book anyway. The second chapter is on conditional expectation, something that is one of the most important concepts needed in understanding stochastic processes. The key thing to keep in mind is that conditional expectation has no explicit formula. Years ago, when I came across this thing I was pretty surprised and was at the same time clueless. If there is no explicit way to compute something, how does one compute then? Slowly after getting my fundas right, I realized that conditional expectation variable needs to be guessed based on two constraints , one that restricts the sigma algebra generated by the variable and the second one involving the expectation of the conditional expectation. The chapter presents a nice set of examples and exercises which make it abundantly clear about the ways to guess the random variable.

The third chapter and fourth chapters talk about Martingales. These are mathematical objects that are essential to understand a whole lot of stuff in math finance and stochastic integration. For example, if you want to integrate a function with respect to Brownian motion, one needs to use Ito’s integral which is nothing but a martingale process. Somehow, I think it is better to read about Martingales from some other book. Its hard to understand the importance of Martingale inequalities just by reading a few pages. The book covers the fundas behind martingales at a blazingly fast speed.

The real fun starts from chapter 5 onwards where Markov chains are introduced. As mentioned earlier, the highlight of this book is “learn by doing”. In that spirit, a basic definition of Markov chain and properties are given and the reader is expected to work out the important properties, lemmas relating to Markov chains. One thing I liked is the application of Fatou’s lemma in proving some lemmas for countable Markov chains. Finally I found some application where I could easily understand the utility of Fatou’s lemma. Chapter 6 is about continuous stochastic processes. Poisson process and Weiner process are covered in this chapter. This is also the chapter where one sees the application of Martingale inequalities.

The last chapter covers Ito’s stochastic calculus. Obviously 30 pages cannot do full justice to Ito’s calculus. But the chapter imparts enough rigor and intuition so that reader gets a good idea of all the important concepts of Ito’s framework. At the very beginning of the chapter, it shows why Ito’s integral is so different from Riemann integral. Subsequently Ito’s integral is constructed for a random step process. These random step processes form the basis for a generalized random process. Also the Ito’s integral for the sequence of random step processes is used to compute the Ito’s integral for a generalized random process with respect to Brownian function. The basic properties of Ito’s integral are stated and derived. Also the sufficient condition for the existence of Ito’s integral is stated and proved. The thing I liked about this chapter is that one gets enough practice in proving two things, one is to formally check whether the hypothesized random step function is a good approximation to a particular integrand function, and second is to verify whether Ito’s integral of the random step process does indeed converge to an Ito process. The chapter ends with a brief discussion of Ito’s lemma and its application in solving stochastic differential equations. Frankly if you are reading this chapter with absolutely no knowledge about Riemann-Stieltjes integrals, it might be a little difficult to put the pieces together.

The book is a good introduction to discrete as well as continuous time stochastic processes. I think the book has two appealing aspects. One is that it has a thorough explanation of conditional expectation and the second is the use of problems as guideposts for the reader to figure out various properties of stochastic processes.