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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.

My question
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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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 '14 at 8:50
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    $\begingroup$ @vonjd Its lecture notes from a course on Mathematical Finance $\endgroup$ – emcor Jul 21 '14 at 9:02
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    $\begingroup$ @emcor Can I have this images in pdf file $\endgroup$ – Zbigniew Feb 23 '15 at 15:49
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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 '17 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 '17 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 '17 at 17:27

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