Almost Sure

Stochastic Calculus

This page is an index into the various stochastic calculus posts on the blog.

Stochastic Calculus Notes

I decided to use this blog to post some notes on stochastic calculus, which I started writing some years ago while learning the subject myself. The aim was to introduce the theory of stochastic integration in as direct and natural way as possible, without losing any of the mathematical rigour. The required background for properly understanding these notes is measure theoretic probability theory. These notes are currently in progress, and are being updated regularly.

Filtrations and Processes

Stochastic Integration

Martingales as Integrators

Special Processes

The General Theory of Semimartingales

The Projection Theorems

Examples and Counterexamples

In addition to the notes listed above, I am also starting to post examples demonstrating the various results and techniques of stochastic calculus, together with counterexamples to show how they can fail if the necessary conditions are not met. In stochastic process theory, in particular, there are often measurability or integrability conditions required which, if they are not met, can cause the expected results to fail in quite subtle ways. The aim is to build up a collection of examples showing what can go wrong, and to help understand the limits of the standard theory.

Brownian Motion

Other Stochastic Calculus Posts

Posts on stochastic calculus which do not fit into the categories above are listed here.


  1. Dear George Lowther,

    I was wondering what your next posts will be talking about. Anything about Local Times, Malliavin Calculus, Quasi-sure Analysis, Backward SDEs, Large Deviations Theory, Stochastic Control Theory ?

    Anyway whatever the subject you may pick I’ll be delighted to read more on your blog,

    Best Regards

    Comment by toultoutim — 26 March 12 @ 9:22 AM | Reply

    • Hi. I already started writing something on quasimartingales, which is a continuation of the “General Theory of Semimartingales”. Actually, I started this before the new year but was busy with other stuff so didn’t get it completed an ready to post. Should post that in a few days. After the general theory then that finishes everything that was originally intended for these notes, although I’ll probably add some stuff such as local times. I’ll also give some thought to the other things you mention, which are quite interesting.

      Comment by George Lowther — 26 March 12 @ 11:43 AM | Reply

  2. Dear Sir,

    First of all, thank you so much for making this blog freely available for all; it will surely help me in my autodidactic pursuit re stochastic mathematics.
    Secondly, I was wondering whether you might help me with something that I have been stuck with and unable to grasp whilst reading the book by Prof Salih Neftci called “An Introduction to the Mathematics of Financial Derivatives”. In there, he gives an equation but I do not know how he got the result. If you can help, then either a) you have the book – I can specify precisely which equation and which page, or b) you don’t have to book: I will handwrite the the equation and set up the problem, scan it and post an upload link as a comment here?


    Comment by WKW — 9 July 12 @ 4:21 PM | Reply

    • Hi WKW,

      Let me recommand you to rather ask this kind of question at the QuantSE forum “” ( or alternatively at forum), where might find a lot of skillfull people on those matters.

      Best regards


      Comment by TheBridge — 10 July 12 @ 8:16 AM | Reply

    • I don’t have that book, but you can try posting it here and I’ll let you know if its something I know. But, as TheBridge mentions, using the QuantSE forum might be a better bet as it will reach a wider audience.

      Comment by George Lowther — 13 July 12 @ 12:00 AM | Reply

  3. I thank u all you all great minds. pls, Ihave to sde model to solve by Ito formula. I will be grateful if any can help with the solution the model is: dXt =[rXt(1-Xt/k)-qzt]dt+gXtdWt

    Comment by Abah Mathias — 12 June 15 @ 10:26 AM | Reply

  4. Notify me of new posts via email.

    Comment by Steven Meyerson — 22 March 16 @ 5:50 AM | Reply

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