I stumbled upon the following problem, I want to compute the stochastic differential of the following 3 processes:
$dX_t = \mu_{X,t}dt + \sigma_{X,t}dW_t$
$dY_t = \mu_{Y,t}dt + \sigma_{Y,t}dW_t$
$dZ_t = \mu_{Z,t}dt + \sigma_{Z,t}dW_t$
That is, I want to calculate $d(X_tY_tZ_t)$ using Ito's product rule
Now from the aforementioned product rule I know that $d(X_tY_t) = X_tdY_t + Y_tdX_t + \sigma_{X,t}\sigma_{Y,t}dt$. I tried to write $P_t = X_tY_t$ and substitute to calculate $d(P_tZ_t)$.
I obtained the process $d(X_tY_tZ_t) = X_tY_tdZ_t + X_tZ_tdY_t + Y_tZ_tdX_t + Z_t\sigma_{X,t}\sigma_{Y,t}dt + \sigma_{P,t}\sigma_{Z,t}dt$ but do not know how to work out the last volatility product. The correct process is apparently equal to
$d(X_tY_tZ_t) = X_tY_tdZ_t + X_tZ_tdY_t + Y_tZ_tdX_t + Z_t\sigma_{X,t}\sigma_{Y,t}dt + X_t\sigma_{Y,t}\sigma_{Z,t}dt+Y_t\sigma_{X,t}\sigma_{Z,t}dt$
I was hoping if somebody could help me arriving at the above equation.
Moreover, I am also curious to find out how to calculate the SDE $d(X_t/Y_t)$, given that the processes $X_t$ and $Y_t$ are the same as given above. If somebody knows how to tackle this one, I would be very grateful for your help.
Many thanks in advance.
Best Answer
I found the answer myself after some trying:
Again, Ito's product rule; $d(X_tY_t) = X_tdY_t + Y_tdX_t + dX_tdY_t$.
Substituting in for $P_t = X_tY_t$, we can write the SDE of the 3 processes as:
\begin{align} d(P_tZ_t) &= P_tdZ_t + Z_tdP_t + dP_tdZ_t\\ \text{(substituting back in ${P_t}$)}&= X_tY_tdZ_t + Z_td(X_tY_t) +d(X_tY_t)dZ_t\\ &= \underbrace{X_tY_tdZ_t + X_tZ_tdY_t + Y_tZ_tdX_t}_{C_t} + Z_tdX_tdY_t + d(X_tY_t)dZ_t\\ &= C_t + Z_tdX_tdY_t + X_tdZ_tdY_t + Y_tdZ_tdX_t + dX_tdY_tdZ_t\\ &=C_t + \sigma_{Y,t}\sigma_{Z,t}X_tdt + \sigma_{X,t}\sigma_{Z,t}Y_tdt + \sigma_{X,t}\sigma_{Y,t}Z_tdt \end{align}
Note: Neglecting all terms smaller than $dt$. However, I am still having trouble with the quotient $d(X_t/Y_t)$. Any help regarding this problem would be much appreciated.