Stochastic Calculus Notes

I have decided to use my blog to post some notes that I initially made on stochastic calculus when learning the subject myself. I wrote these after reading through some books which took an unnecessarily long and difficult route to get to the interesting stuff which I was interested in. Complicated and rather obscure subjects such as optional and predictable projection and a lot of theory of continuous-time martingales were dealt with at length before getting round to the general theory of stochastic integration. Consequently, I decided to go through the theory myself in a more direct way, while still working out rigorous proofs of all the more useful theorems which I was interested in learning. The result was three small notepads containing the following.

  • Basic definitions regarding continuous-time filtrations, adapted processes, predictable processes, stopping times, martingales, etc.
  • Some useful elementary results such as the debut theorem for right continuous processes and the existence of cadlag versions of martingales
  • Definition of stochastic integration and elementary properties.
  • Definition and elementary properties of quadratic variations.
  • Ito’s formula, including the generalized Ito formula for non-continuous processes.
  • Stochastic integration with respect to martingales.
  • The Doob-Meyer decomposition.
  • Quasimartingale decompositions.
  • Decompositions of semimartingales.
  • Decompositions and integration with respect to special semimartingales.

This covers a lot of the general underlying theory required. Of course, being able to apply this to practical applications requires further knowledge of stuff like stochastic differential equations. Time permitting, I’ll start to post these notes here.As I have learned much more since originally making these notes, I will attempt to simplify or improve on the originals where possible.

The prerequisite knowledge required to properly understand these notes is measure theoretic probability theory (e.g., properties of the Lebesgue integral such as dominated convergence, Fubini’s theorem, L^p spaces, convergence in probability, etc.).

9 thoughts on “Stochastic Calculus Notes

  1. Mr. Lowther,

    Your exposition is very clear and to the point.
    Are your notes available in a pdf or similar format?

  2. H Baw – Thanks for your comment.

    No, unfortunately, they’re not. At the moment all I have is some handwritten notes I made a few years ago, and am currently writing up in stages and posting on here.

    I’m typing these up in LaTeX and converting to wordpress posts, so I could join them all together when it’s done to create a single big pdf.

  3. Hello,
    Thnak you very much for your reply.
    This is somewhat of a personal request, and may
    not be suitable for posting here. But, then there is
    no other way of interacting with you…I have
    a background in basic Measure Theory and Probability Theory (e.g., at the level of Durrett’s
    text). I wish to learn about Stochastic Calculus,
    but there appear to be many books — each with
    own philosphy, and all require you to survive a
    very demanding preamble/setting up of machinery.
    The integration of Discontinuous Processes appears
    to be the hardest. I have been reading your superb
    blog, but do you have any advice on how one
    might proceed?
    Kind Regards…

    1. That seems to be a perfectly good question for posting here. I agree, many stochastic calculus books do approach the subject from their own specific viewpoint and can demand a lot of setting up of machinery. I also think it can be bit hard to “see the wood from the trees”. However, a rigorous and approach which covers most of the important theory is always going to be a bit tricky to approach. On the other hand, books aimed at specific applications (e.g., finance) approach the useful stuff much more directly, but you have to take the mathematics on faith.

      For a decent and mathematically rigorous approach to the subject, I think Protter is very good and a very readable text. Rogers and Williams (vol 2) is pretty good too, although there is a large diversion on stochastic calculus on manifolds which you probably don’t need.

      Merry Christmas!

  4. btw, with my previous post on local martingales, I have completed the “filtrations and processes” part of my notes, which completes the definitions and useful elementary results.

    Moving on to stochastic integration next. The next post will define the stochastic integral, and I’ll follow that up by stuff such as Ito’s formula. This is using my own approach to the subject, which is a bit different from any textbooks I know (but we end up at the same place in the end). If you have any questions don’t hesitate to ask.

    With Christmas, might not get the chance to post this for a few days though.

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