I am currently taking a finance course which includes some math that is currently above my level, it is however not a pure math class and we are just supposed to be able to apply the math to the given problems without much understanding.
The problem is the following:
We have two stochastic variables $X$ and $Y$, with $Z$ = $X$ $*$ $Y$.
We have the following equations $dX = \mu_{1} X dt + \sigma_{1} X dB_{1}(t)$ and $dY = \mu_{2} Y dt + \sigma_{2} Y dB_{2}(t)$ where $dB_(t)$ is a Wiener process.
The exercise is to calculate $dZ$.
I suppose I should apply Ito's lemma in some way but I am not sure how to approach this.
Thanks in advance!
You know that $d\langle X,Y\rangle (t) = \sigma_1 X(t) \times \sigma_2 Y(t) \times d\langle B_1, B_2\rangle (t) $.
According to your question, I don't know what is the link between $B_1$ and $B_2$. Then I will assume that $(B_1, B_2)$ is a vectorial gaussian process and that the correlation $\rho(B_1(t), B_2(t)) = \rho$ does not depend on $t$ (for instance, if $(B_1, B_2)$ is a Brownian motion then $\rho = 0$; if $B_1 = B_2$ then $\rho = 1$).
In that case, $d\langle B_1, B_2\rangle (t) \rho dt$. Now you are ready for the Ito formula: let $F(x,y) = xy$. Then,
\begin{align} dZ(t) &= dF(X(t),Y(t)) = \frac{\partial F}{\partial x}(X(t), Y(t)) dX(t) + \frac{\partial F}{\partial y}(X(t), Y(t)) dY(t) \\&+ \frac 12\left ( \frac{\partial^2 F}{\partial x^2}(X(t), Y(t)) d\langle X,X\rangle (t) + 2 \frac{\partial^2 F}{\partial x \partial y}(X(t), Y(t)) d\langle X,Y\rangle (t) + \frac{\partial^2 F}{\partial y^2}(X(t), Y(t)) d\langle Y,Y\rangle (t) \right) \\&=Y(t) dX(t) + X(t) dY(t) + \sigma_1 \sigma_2 X(t) Y(t) d\langle B_1, B_2\rangle (t) \end{align}
This is the general product formula. Here it simplifies more: $$ = X(t)Y(t)\left[(\mu_1 + \mu_2 + \rho \sigma_1 \sigma_2)) dt + \sigma_1 dB_1(t) + \sigma_2 dB_2(t) \right] $$
If you just want to remember something useful, just remember the formula $$ dX(t) = b_1(t) dt + c_1(t) dB_1(t), dY(t) = b_2(t) dt + c_2(t) dB_2(t)\\ \implies d(XY)(t) = X(t)dY(t) + Y(t) dX(t) + \rho c_1(t) c_2(t) dt $$ with the hypothesis given earlier about $B_{1,2}$.