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I wish to study stochastic processes in my free time. Since this is just as a hobby, I wanted to go with a cheap textbook. In my endeavor I have found four candidates within my budget of 40$.

I have the basics of statistics and probability but a much stronger calculus background to go with it (Tensor calculus, etc.). My interests are more theoretical than practical and I won't shy away from mathematical formalism, rather that's where my curiosity lies. I would like a "medium" level of mathematical rigor, providing satisfying justifications but not getting bogged down too hard into the details.

I would like to know which of these might be recommended and why?

  1. Introduction to the Theory of Random Processes by Gikhman & Skorokhod

  2. Stochastic Processes and Filtering Theory by Jazwinski

  3. An Introduction to Probability and Stochastic Processes by Melsa & Sage

  4. Stochastic Processes by Parzen

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I would say 'Essentials of Stochastic Processes' is a good introductory book (and it is available on the author's website as it gives the properties of Poisson processes, markov chains, and martingale theory.

You could also look at an Introduction to stochastic integration by Kuo. What I really like about this book is that it gives the definition of the stieltjes integral and motivates the fact that to integrate against a stochastic process, we must change the definition of the integral. In the second chapter or so, it gives the fundamentals of the Brownian motion, one of the most studied continuous time stochastic processes.

In this route, the book by Oksendal is also recommendend (and it also includes a chapter on filtering, which you are also looking into by your references).

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