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In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes equation are given.

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I am looking for references where lots of worked examples of applying Ito's lemma are given in an easy to follow, step by step fashion. Also more advanced cases should be covered.

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    $\begingroup$ This post has been up for a while, is great, and helped me through understanding this topic more. I also wanted to update it with additional resources that I found → math.drexel.edu/~song/Gene%20Golub%20Summer%20School/Song/… $\endgroup$
    – Chef1075
    Jun 19, 2019 at 18:59
  • $\begingroup$ @Chef1075: Thank you for those additional resources. $\endgroup$
    – vonjd
    Jun 26, 2019 at 11:39

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These are all examples on Ito Formula in its general form (with quadratic variations):

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    $\begingroup$ +1: Thank you, could you please also give a source $\endgroup$
    – vonjd
    Jul 21, 2014 at 8:50
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    $\begingroup$ @vonjd Its lecture notes from a course on Mathematical Finance $\endgroup$
    – emcor
    Jul 21, 2014 at 9:02
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    $\begingroup$ @emcor Can I have this images in pdf file $\endgroup$
    – Zbigniew
    Feb 23, 2015 at 15:49
  • $\begingroup$ @emcor Would be able to provide the URL to the lecture notes from this course? $\endgroup$
    – Jessie
    May 29 at 20:04
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I thought this was an interesting example to add. It concerns a "ratio model" of habit (as opposed to a "difference" model of habit). See, for example, Abel (1990, American Economic Review). Let $$ x_t = \lambda \int_{-\infty}^t e^{-\lambda(t-s)} c_s ds. $$ (For context, $x_t$ is a log habit index that is given by a geometric average of past consumption, where $c_t$ is log consumption.) Then by Ito's formula, \begin{align} d x_t &= \lambda \int_{-\infty}^t -\lambda e^{-\lambda(t-s)} c_s ds \, dt + \lambda c_t dt \\ &= \lambda (c_t - x_t) dt. \end{align} The part that is interesting to me is the that it easy to err in thinking that the answer is $dx_t = \lambda c_t dt$ or $d x_t = -\lambda x_t dt$.

EDIT: Here, $c_s$ is some well-behaved stochastic process. This is essentially the same as 9-1 (a) above when $dc_t = dW_t$, where $W$ is a Brownian motion. This kind of calculation seems to show up somewhat frequently (Hull-White interest rate model), but doesn't seem to directly use Ito's lemma.

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  • $\begingroup$ Where do you need Ito's lemma in this? Looks like plain Leibniz rule to me - en.wikipedia.org/wiki/Leibniz_integral_rule#Formal_statement. $\endgroup$
    – LocalVolatility
    Feb 13, 2017 at 8:08
  • $\begingroup$ @LocalVolatility I've edited the answer to reflect your comment. I guess the point is that $c_t$ can be a stochastic process and the calculation still works out the same (example 9-1 (a) above). However, I can remove the answer entirely if you think that's best. $\endgroup$
    – jmbejara
    Feb 13, 2017 at 17:05
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    $\begingroup$ I think there is no harm in keeping it as your edit makes it clear that it doesn't directly invoke Ito's lemma. $\endgroup$
    – LocalVolatility
    Feb 13, 2017 at 17:27

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