# Second order derivatives in Ito formula for Brownian motion and local martingale

Itô's formula for a $$\mathcal{C}^2$$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}(X_s, Y_s) dX_s \\ &+ \frac{1}{2}\int_0^t\frac{\partial^2 F}{dx^2}(X_s, Y_s) d\langle Y, Y\rangle_s + \frac{1}{2}\int_0^t \frac{\partial^2 F}{dy^2}(X_s, Y_s) d\langle X, X\rangle_s \\ &+ \frac{1}{2}\int_0^t \frac{\partial^2 F}{dx \,dy}(X_s, Y_s) d\langle X, Y\rangle_s \end{align}

However, it seems that when applied to 1) $$X_t = t$$ and $$Y_t = B_t$$ (Brownian motion in 1D) , or 2) $$X_t$$ local martingale and $$Y_t = \langle X, X \rangle_t$$, the last two integrals disappear in the formulas I've seen.

Why is that so ?

• Because the quadratic covariation $\langle f, g \rangle$ equals zero if $f$ is continuous and $g$ of bounded variation – saz Feb 27 at 13:00
• Because $dtdB_t = o(t^{3/2})$ and $(dB_t)^2 = dt$. In the first case when, you expand the terms $d[X,X]_s$ and $d[X,Y]_s$, you get things which are $o(ds^\alpha)$ with $\alpha>1$ and so can be thrown away. – dohmatob Feb 27 at 15:28
• – dohmatob Feb 27 at 15:34
• Thank you, @dohmatob I think that using small o notation here is somewhat of an abuse, and also does not explain the result for the martingale. But it's still useful as mnemonics! – P. Camilleri Mar 1 at 5:23
• My comment was for the first of your cases. For the second, you'll need to provide more details / refs. It's usually best to post questions which are self-contained, in terms of terminology, etc. – dohmatob Mar 1 at 10:18