Application of Ito's formula

Question

I have the following process:

\begin{equation*} X_t= \exp \left(\int_{0}^{t}s \, dB_s-\frac{t^3}{6} \right), \end{equation*}

where $B$ is a Browinan motion.

My textbook asks to write Ito's formula for $X$ and show that $X$ is a martingale.

I don't really know what the excercise wants, shouldn't I have some function so that I can write Ito's formula?

Answer 1

For an Itô process $(X_t)_{t \geq 0}$ of the form

$$X_t-X_0 = \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds \tag{1}$$

Itô's formula reads

$$f(X_t)-f(X_0) = \int_0^t f'(X_s) \sigma(s) \, dB_s + \int_0^t \left( \frac{1}{2} f''(X_s) \sigma^2(s) + f'(X_s) b(s) \right) \, ds.$$

Now:

1. $X_t := \int_0^t s \, dB_s - \frac{t^3}{6}$ is an Itô process. Choose $b$, $\sigma$ such that the representation $(1)$ holds.
2. Apply Itô's formula for $f(x) := \exp(x)$.