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Notes & exercises MATH4/67201 Martingale Theory
Preface
This “Quarto book” contains all notes, exercises and solutions for the course MATH4/67201 Martingale Theory at the University of Manchester. For course details & other resources, see the course page at Blackboard.
- Every chapter has a section with exercises, their solutions will become available (underneath the exercises) after the Friday tutorial in which we discussed them (refresh the page if needs be).
- This material comes only in this format, no pdf’s or other formats I’m afraid (yet — maybe in the future if there is demand for it and I have had the time to sit down and get these on par as well). If this is annoying to you because it disturbs your usual workflow then I can at this point only apologise. You could consider using the “Print to pdf” functionality of your web browser to still produce your own pdf’s — from my quick testing it seems that Chrome/Chromium does the most satisfactory job with this. Further there are annotation tools that can handle web based materials (if you like using one and your favourite one can’t), see e.g. this and this thread on Reddit for some inspiration.
- At the moment (September 2024), this material is still a work in progress — updates will follow as we make our way through the semester!
Acknowledgements & further/other reading
It goes (almost) without saying that these materials rely heavily on some of the (imho) seminal textbooks in the field, in particular (full list of references/details here):
- “Probability and random processes” by Grimmett and Stirzaker,
- “A Concise Introduction to the Theory of Integration” by Stroock,
- “Probability with martingales” by Williams,
- “Foundations of Modern Probability” by Kallenberg.
Of these, Grimmett and Stirzaker’s book focuses mostly on relatively elementary probability theory with some exploration of stochastic processes while Stroock’s book deals with the basics of measure theory (in particular, the Lebesgue measure & Lebesgue integration). These two books are brilliant as a “first course” in their respective topics and are probably most accessible/easiest to read. Williams’ book, though also covering probability theory and measure theory basics to an extent, is mainly concerned with the development of the theory of martingales (in discrete time). It probably comes next in terms of accessibility (but still makes for a very good read!). Finally Kallenberg’s book covers the previously mentioned topics as well plus a lot more fascinating material in the field of probability theory & stochastic processes. I’d guess that most of you would consider this one a bit harder to read than the other ones. All these books are available via the UoM library.
Of course, all (examinable) materials are covered in these notes so from that perspective you shouldn’t worry about needing any extra resources but in case you would like something else/extra, these books are a great place to look.
Notice any issues/mistakes?
Obviously the above comes with the qualifier that any mistakes/typos/issues/unclarities/… are completely my own responsibility. If you catch any then I’d be very grateful if you would let me know (email address by clicking on my name at the top of this page), thank you!