I would like to find a book that introduces me gently to the subject of stochastic processes without sacrificing mathematical rigor. It would be great if the book has lots of examples and that the book is designed for undergraduates. Just like how there are rigorous undergraduate abstract algebra books (I am thinking of GAllian's contemporary modern algebra here).
I already studied measure theory, and prob. theory.
Thank you.
Here is my own (admittedly, personally biased) list. I do not claim it is better than anyone else's list, but at least I do know them all very well, having taught undergraduate stochastic processes courses out of each in various decades.
All are mathematically rigorous, and all are below the measure theoretic level. A full dose of lower-division calculus and perhaps a beginning course in probability would be helpful background. All of the authors have clearly used much of the material in real applications, which I think is important for beginners at the undergraduate level. Listed in order of publication.
1) Appropriate parts of Feller (vol. 1): A classic book with many surprisingly difficult problems, but heavy on intuitive explanations.
2) Parzen: Somewhat unusual approach and collection of topics, but based on real-world concerns, and with intuitive explanations. (Rigorous enough for many years of use at Stanford.)
3) N.T.J. Bailey: Many examples from biological sciences, but mathematically rigorous. Includes time-continuous processes.
4) Roe Goodman: Meticulously clear, often intuitive. Some attention to simulation, but not as a substitute for rigorous presentation. More attention to queueing and other time-continuous processes than in many books accessible to undergraduates.