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

\begin{equation} Z(t) = \exp (a W(t)) \end{equation}

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. – LocalVolatility Dec 9 '16 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 '16 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. – LocalVolatility Dec 9 '16 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 '16 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 '16 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 $dZ_t=df(W_t)$.