Applying Itô’s Formula to Exponential Moving-Average Habit
Summary
The document asks for step-by-step references on applying Itô’s lemma beyond familiar geometric Brownian motion and Black–Scholes examples. The visible answers provide little reference material, but one adds a worked calculation for a log habit index defined as an exponentially weighted average of past log consumption. Differentiating the integral gives a drift that depends on the difference between current consumption and the current habit index.
The example highlights why both the changing weight on past observations and the contribution from current consumption matter: omitting either term gives an incomplete dynamic equation. The answer notes that similar calculations appear in settings such as the Hull–White interest-rate model, while acknowledging that the calculation itself does not directly rely on Itô’s lemma in the usual sense. The document does not supply the broader collection of advanced worked examples sought by the original question, so its instructional coverage is narrow.
Key ideas
- An exponentially weighted average of past consumption has dynamics driven by current consumption relative to the habit index.
- The integral’s changing exponential weights contribute a decay term to its derivative.
- The current observation contributes separately from the weighted history.
- The example illustrates a useful stochastic-process calculation but does not provide a broad set of advanced Itô examples.
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# Worked examples of applying Ito's lemma
# 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.
## Answer by emcor (score 29, accepted)
https://quant.stackexchange.com/a/14104
These are all examples on Ito Formula in its general form (with quadratic variations):
## Answer by jmbejara (score 6)
https://quant.stackexchange.com/a/16781
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.