# How to define the $f$ function to apply Ito's lemma?

$$Z(t) = \exp (a W(t))$$

I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $dZ$ function and not the other way around.

My question: I want to use Ito's lemma to find $dZ$. How do I define my $f$ (from the standard Ito's lemma formulation) function?

• With a bit of effort you should find this in pretty much any textbook. I random pickup up Volume II of Shreve's Stochastic Calculus for Finance. The first worked example in Section 4.4.3 is pretty much already what you are looking for. Dec 9, 2016 at 15:10
• I don't have that book. I have taken this example from Bjork's book. But I still have trouble understanding how to define the "f" function
– user25295
Dec 9, 2016 at 15:14
• Your self-study tag contradicts your own statement that this is homework. I am voting to close this question as too basic. You should be able to infer it from the worked-out Example 4.13 in Bjoerk's book. Dec 9, 2016 at 15:34
• In that example Bjork sets sigma=1 and mu=0 ( standard Ito's lemma formulation ). I don't quite follow that logic.
– user25295
Dec 9, 2016 at 15:40
• It is a basic question, but basic stochastic calculus not basic finance strictly speaking. I think it can serve the community. But no offense @LocalVolatility, it is a really borderline question. I hope you won't mind.
– SRKX
Dec 9, 2016 at 15:46

In fact, the variable $Z_t$ is a function of $W_t$, which is the stochastic variable.
Therefore, you can see $Z_t$ as $f(W_t) = \exp(aW_t)$.
The rest is a trivial application of Ito's lemma to find $dZ_t=df(W_t)$.