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In "Baxter, Rennie" book, there is an explanation of product rule for Ito formula. They apply Ito formula (without any details) to

$$\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2) = X_tY_t$$

and obtain

$$d(X_tY_t) = X_t\,dY_t + Y_t\,dX_t + dX_t\,dY_t$$

There is a good expalantion how to get $d(X_tY_t) = X_t\,dY_t + Y_t\,dX_t + dX_t\,dY_t$ in "Wiersema" by applying Ito formula, but he doesn't use $\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2)$ at all.

Could you please explain this step? I don't see how applying Ito lemma to $\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2)$, I would come up with result.

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  • $\begingroup$ May I know how 'Wiersema' applies Ito formula to obtain the product rule? $\endgroup$
    – Idonknow
    Commented Oct 19, 2019 at 14:51
  • $\begingroup$ You can find it in the book books.google.ch/… $\endgroup$ Commented Oct 21, 2019 at 10:29

1 Answer 1

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1.) Let $Z_t = X_t^2 = f(X_t)$ then Itô's Formular leads to:

$$dZ_t = f'(X_t)dX_t + \frac{1}{2}f''(X_t)(dX_t)^2 = 2X_tdX_t + (dX_t)^2$$

2.) $$d(X_tY_t) = d\left(\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2)\right) = \frac{1}{2}\left(d(X_t + Y_t)^2 - dX_t^2 - dY_t^2\right)$$

Now applying 1.) on all three parts let us realize:

1.) $$\begin{align*} d(X_t+Y_t)^2 &= 2(X_t+Y_t)d(X_t+Y_t) + \left(d(X_t + Y_t)\right)^2 \\ &= 2\left(X_tdX_t + X_tdY_t + Y_tdY_t + Y_tdX_t) + (dX_t)^2 + 2dX_tdY_t + (dY_t)^2\right) \\\\ dX_t^2 &= 2X_tdX_t + (dX_t)^2 \\ dY_t^2 &= 2Y_tdY_t + (dY_t)^2 \end{align*}$$

Plug in in 2.): $$d(X_tY_t) = \frac{1}{2}\left(2X_tdY_t + 2Y_tdX_t + 2dX_tdY_t\right) = dX_tdY_t + Y_tdX_t + dX_tdY_t$$

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