Itô's formula and Integration by parts

Question

Could someone please help with this question:

By applying the generalized Itô’s formula to the 2-dimensional process $ {\{(Xt,Yt),t \ge 0 }\}$ with the function $ F(x,y) = xy $, show the integration by parts formula

$$ X_{t}Y_{t} = X_{0}Y_{0} + \int_{0}^t X_{s}dY_{s} + \int_{0}^t Y_{s}dX_{s} + VAR[X,Y]_{t} $$

assuming the necessary integrability conditions which are necessary to make sense of the previous formula. In which case we get the usual formula for the derivative of the product?

Thanks in advance!

Answer 1

This is almost direct from the definition of Itô's formula. The partial derivatives are $F_x = y$, $F_y = x$, $F_{xx} = F_{yy} = 0$, $F_{xy} = 1$. So the formula reads: \begin{equation} X_{t}Y_{t} = X_{0}Y_{0} + \int_{0}^t F_{x}(X_s, Y_s)dX_{s} + \int_{0}^t F_{y}(X_s, Y_s)dY_{s} + \int_{0}^t F_{xy}(X_s, Y_s)d[X,Y]_{s} = \\ = X_{0}Y_{0} + \int_{0}^t X_{s}dY_{s} + \int_{0}^t Y_{s}dX_{s} + [X,Y]_{t}. \end{equation}