I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal.

So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on probability with measure theory and some courses applying those concepts.

The probability reference was some chapters of Probability with Martingales by David Williams.

Where do I go from here? What textbooks do you recommend?

I believe I lack knowledge on a lot of the basics such as, different types of convergence of random variables, laws of large numbers, Malliavin Calculus (and the calculus of variations), Radon-Nikodym stuff, proofs of basic stochastic calculus results (like Girsanov theorem and Ito's lemma), etc.

I'm also looking up recent journal publications. Know where I can look?

Please provide feedback if you think the question can be improved. Any help is appreciated. I don't mind anyone posting comments as answers so I can upvote, I guess.

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    $\begingroup$ Stochastic processes and their applications used to be a top journal in the field. If you pick articles from it, then research where else their authors publish, you probably get many/most good ones - if things haven't changed completely in the last 10 years. $\endgroup$ – gnometorule Apr 4 '15 at 13:37
  • $\begingroup$ @gnometorule 1 thanks for commenting even though this already had an accepted answer. How did you find this? $\endgroup$ – user198044 Apr 9 '15 at 15:47
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    $\begingroup$ You're welcome! I think because I read your thread of questions about stochastic analysis/calculus. Had a reply on one of those too, but deleted it as it seemed of no use. ;) (it's a valid question, and I don't understand why it was received poorly) $\endgroup$ – gnometorule Apr 9 '15 at 17:40
  • $\begingroup$ @gnometorule Awww thanks :) $\endgroup$ – user198044 Apr 10 '15 at 8:26

just to offer two cents on this (health warning is that this is from an ex-trader rather than quant, and based on personal experience through self-study rather than eg as part of a formal PhD programme):

Williams really is fantastic, learned the basics of measure theoretic probability from that as an undergrad, and it's stood the test of time and is still a classic.

After this, some natural canonical texts would be:

  1. Rogers and Williams' two volumes: Diffusions, Markov Processes and Martingales

I think this really does contain a huge amount of material (in addition to containing the material from Probability with Martingales so in some sense is a natural transition). You'll certainly find the standard Ito/Girsanov/Radon-Nikodym material well presented therein.

A different tack would be:

  1. Oksendal: Stochastic Differential Equations,

This was very popular when I was reading up on SDEs, and has a somewhat less formal style than some of the other standard references.

  1. Karatzas and Shreve: Brownian Motion and Stochastic Calculus

This is a bit more encyclopaedic than Oksendal, but again was very popular when I was reading the material about 10 or so years ago. More heavy going than Oksendal, and possibly overkill if the ultimate aim is more finance than analysis orientated.

  1. Revuz and Yor: Continuous Martingales and Brownian Motion

I didn't use this myself, (again I was reading for interest and as ancillary to finance rather than for embarking on a stochastic analysis PhD - am a number-theory/algebra nut at heart!), but this is I think a classic text, although more formal than the others I've mentioned.

Finally a rather pleasing book is:

  1. Bobrowski: Functional Analysis for Probability and Stochastic Processes

This has a nice survey, as the title suggests, of some of the functional analytic underpinnings of measure-theoretic probability, and I found the exposition a delight to read.

Hope some of those help, these are not finance books (sounds like you've got that covered), definitely can offer some views on that side of things should you need depending on which area of finance might interest most (credit/rates etc..). Many of the finance books by authors such as Brigo are highly rigorous but much much better suited to assimilating the finance concepts and acquiring facility with actual problems that matter 'at the coal face', but again depends on the aim / perspective.

Good luck and cheers!

  • $\begingroup$ Thanks. 1. Any idea where to look for recent journal publications? (I guess I wouldn't be able to understand them until I read the books you suggested, but just so I know where to look) 2. Cool. You read those links within links hehehehe. I have no plans of going back into finance for now. Thanks MathVandal! I really appreciate this! $\endgroup$ – user198044 Mar 25 '15 at 8:20
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    $\begingroup$ Hey there - sadly am definitely not an expert on where to find good journal publications on stoch. analysis. There are probably many on this forum amply qualified to help you, else you could always try math.overflow? Other than that how about looking where Martin Hairer, one of last year's Fields' Medallists has been publishing ;-) ! $\endgroup$ – Mehness Mar 25 '15 at 14:09
  • $\begingroup$ It's off-topic there. Oh well. Thanks anyway. $\endgroup$ – user198044 Mar 27 '15 at 17:53
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    $\begingroup$ Oh sorry bout that - am a newbie still learning the ropes... $\endgroup$ – Mehness Mar 27 '15 at 18:22
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    $\begingroup$ Hi @JackBauer - pretty tough to give a hard and fast rule re overlapping topics honestly. What I tended to find myself doing is using one book as the main bedrock, then perhaps another as 'reading material'. And certainly curating exercises between them. I do love Williams' style for example, so would probably use Rogers and Williams as 'core' and perhaps Oksendahl for a lighter read and further exercises. It's nice to get a variety of styles but that I guess only if you have the luxury of time which you may not! $\endgroup$ – Mehness Sep 4 '18 at 13:39

I think it takes courage to identify what you are weak at and make that public. The mindset of identify weaknesses and seeking to improve is essential. If you is able to keep this mindset awake even when struggling, progress will be but a mere trivial corollary to the rich understanding that you will gain.

Here are some thoughts, based on my experience:

a) I remember (painful memories are usually recalled quickest) when I didn't know what a Radon-Nikodym derivative was. Nor did I really understand why we care to study martingales. At times studying this stuff was (and still can be) torture.

Fortunately, I now do understand. How did I get here? By studying and really trying to understand what the authors are saying. Discussions with fellow interested peers helped. Computer simulations helped.

b) Stochastic analysis is a subset of probability theory, a field of mathematics created to satisfy gamblers. If you are into gambling, stochastic analysis will make sense. Physicists also love stochastic calculus, but for different reasons.

For example, topics such as 'hitting times' will be obvious to any (good) gambler - the gambler wants to know what level he should hit before he can leave the casino.

There are a lot of good books, non mathematical nature, that give context on the history of stochastic analysis and how it came to dominate finance. Physicists on Wall Street by some bloke (a physicist) is pretty good.

c) The books listed by MathVandal are fantastic. Williams's book is superb: at the start he has the (successful) candour to explain the purpose of measure theory and topological spaces in a few lines. There are many other books worth reading: Schilling has a book on martingales and measure theory, Kreyszig has a book on functional analysis and Klebaner has a book on stochastic calculus. I despise many mathematical finance books, but anything by Wilmott is pretty good.

d) If I were you, I would not be punishing myself if you do not understand the Malliavin calculus. It is a very difficult topic. Yes, it is just the calculus of variations applied to stochastic processes on some space - that is trivial, when you think about it. But Malliavin's idea is hard to express - there are many spaces that are degenerate in some sense and Malliavin invents new tools that attempt to tackle these degeneracies. For example, he invents 'quasi surely' as opposed to almost surely to refine behaviour of paths. It takes a lot of time to understand this as opposed to the plethora of symbols and theorems that are thrown at you.

  • $\begingroup$ Thanks Arbias. As for letter d, I just realized that Malliavin Calculus is just one branch (what's the term?) of stochastic calculus. Ito Calculus is the other, so I guess I'll stick to that. $\endgroup$ – user198044 Apr 16 '15 at 12:42
  • $\begingroup$ Arbias, is this it? "Measures, Integrals and Martingales: René L. Schilling" $\endgroup$ – user198044 Apr 16 '15 at 16:26
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    $\begingroup$ It depends on what you define as 'calculus'. A stochastic calculus can be defined, simply speaking, as the useful tools and results that follow given a definition of a stochastic integral. In this sense, Malliavin calculus is different to Ito calculus because the former operates under the Skohorod integral whilst the latter operates under the Ito integral. It can be shown that sometimes the Skohorod integral and Ito integral coincide and have the same value, therefore Malliavin calculus and Ito calculus are actually connected to each other! $\endgroup$ – AXH Apr 18 '15 at 13:46
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    $\begingroup$ Yes, that is the book. Schilling has another book on Brownian Motion, which is essentially a guide to stochastic calculus. Also, Schilling provides solution manuals (to the exercises contained in his books) on his website, somewhere.. They are quite easily accessible. $\endgroup$ – AXH Apr 18 '15 at 13:49
  • $\begingroup$ So Arbias, Stratonovich Integral could have or has its own calculus? What about for regular non-stochastic integrals? Do Riemann, Lebesgue, Darboux, Kurzweil-Henstock Integrals have their own calculus? $\endgroup$ – user198044 Jul 29 '15 at 16:13

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