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

Point Processes

Local Times

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.

12 thoughts on “Stochastic Calculus

  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

    1. 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.

  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?


    1. 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


    2. 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.

  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

  4. Hi there, guys I need an urgent help for this exercice. May you help me ont this ?
    Thanks in advance ?

    A continuous random variable Y with a support[0,∞) is said to be exponentially distributed with a parameterλ>0 if its cumulative distributive function is

    Compute the expectation and variance of an exponentially distributed random variable.
    Prove that an exponentially distributed radom variable Y belongs to the functional space
    ,i.e., that E[∣∣Yn∣∣]<∞. Give a expression for E[Yn].

    1. Hi Sacha,

      There is a well known result tha says the following:

      For every nonnegative random variable Y defined on a probability space (\Omega,\mathcal{F},\mathbb{P}) and a real number p > 0 the next equality is true, \int_{\Omega} Y^p d\mathbb{P}(\omega) = \int_{0}^{+\infty} p t^{p - 1} \mathbb{P}(Y \geq t) d\lambda(t).

      With this at hand and the usual formula Var(Y) = \mathbb{E}(Y^2) - \mathbb{E}(Y)^2 you can answer every thing that you ask.

    1. I did once try compiling the entire set of notes into a single pdf, but it is a bit tricky, especially getting all the links correct, so didn’t seem worth the time.
      I do write these posts in latex before converting to wordpress format, so have pdf files locally. Not sure if there is a simple way of making them all available.

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