# Worked examples of applying Ito's lemma

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.

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

• +1: Thank you, could you please also give a source – vonjd Jul 21 '14 at 8:50
• @vonjd Its lecture notes from a course on Mathematical Finance – emcor Jul 21 '14 at 9:02
• @emcor Can I have this images in pdf file – Zbigniew Feb 23 '15 at 15:49

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.

• 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. – LocalVolatility Feb 13 '17 at 8:08
• @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. – jmbejara Feb 13 '17 at 17:05
• I think there is no harm in keeping it as your edit makes it clear that it doesn't directly invoke Ito's lemma. – LocalVolatility Feb 13 '17 at 17:27