Applying Ito’s Lemma to an Exponential of Brownian Motion

Summary

The question asks how to choose the function in Ito’s lemma to find the differential of a process defined as the exponential of a constant times Brownian motion. The answer clarifies that the process is a function of the underlying stochastic variable, Brownian motion, so the appropriate function is the exponential mapping from that variable to the process.

Ito’s lemma is then applied to this function of Brownian motion to obtain the process differential. The exchange explains how to set up the substitution, but does not write out the resulting differential or discuss applications such as pricing, simulation, or model assumptions. It is a concise illustration of identifying the state variable and function before using the lemma.

Key ideas

• The exponential process is treated as a function of Brownian motion.
• In Ito’s lemma, identify the stochastic variable first and express the target process as a function of it.
• Applying Ito’s lemma to that function yields the differential of the process.
• The exchange gives the setup but does not show the expanded differential.

Tags

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


# 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?

## Answer by SRKX (score 3, accepted)

https://quant.stackexchange.com/a/31420

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 $dZ_t=df(W_t)$.

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