Let $X=(X^1,\ldots,X^n)$ be a continuous semimartingale and let $b: \mathbb{R}^{n-1} \to \mathbb{R}$ be a continuous function such that the process $b^X = b(X^1,\ldots,X^{n-1})$ is a semimartingale. Setting $C=\{(x_1,\ldots,x_n)\in \mathbb{R}^n | x_n < b(x_1,\ldots,x_{n-1})\}$ and $D=\{(x_1,\ldots,x_n)\in \mathbb{R}^n| x_n > b(x_1,\ldots,x_{n-1})\}$ suppose that a continuous function $F: \mathbb{R}^n \to \mathbb{R}$ is given such that $F$ is $C^{i_1,\ldots,i_n}$ on $\bar{C}$ and $F$ is $C^{i_1,\ldots,i_n}$ on $\bar{D}$ where each $i_k$ equals 1 or 2 depending on whether $X^k$ is of bound variation or not. Then the following change-of-variable formula holds:
$\begin{eqnarray}F(X_t)&=& F(X_0) + \Bigsum_{i=1}^n \Bigint_0^t \frac{1}{2} \( \frac{ \partial F }{ \partial x_i } (X_s^1,\ldots,X_s^n+) + \frac{ \partial F }{ \partial x_i } (X_s^1,\ldots,X_s^n-)\)\, dX_s^i \\ && + \Bigsum_{i,j=1}^n \Bigint_0^t \frac{1}{2} \( \frac{ \partial^2 F }{ \partial x_i \partial x_j } (X_s^1,\ldots,X_s^n+) + \frac{ \partial^2 F }{ \partial x_i \partial x_j } (X_s^1,\ldots,X_s^n-)\) \,d\langle X^i,X^j \rangle_s \\ && + \Bigint_0^t \frac{1}{2} \( \frac{ \partial F }{ \partial x_n } (X_s^1,\ldots,X_s^n+) - \frac{ \partial F }{ \partial x_n } (X_s^1,\ldots,X_s^n-)\) I(X_s^n=b_s^X) \,d\ell_s^b(X)\end{eqnarray}$
where $\ell_s^b(X)$ is the local time of $X$ on the surface $b$ given by:
$\ell_s^b(X) = \mathbb{P} - \lim_{\epsilon\downarrow0} \frac{1}{2\epsilon} \Bigint_0^s I(-\epsilon < X_r^n-b_r^X < \epsilon )\, d\langle X^n-b^X,X^n-b^X\rangle_r$
and $d\ell_s^b(X)$ refers to the integration with respect to $s\mapsto \ell_s^b(X)$. The analogues formula extends to general semimartingales $X$ and $b^X$ as well. A version of the same formula under weaker conditions on $F$ is derived for the semimartingale $(t,X_t,S_t)_{t\geq0}$ where $(X_t)_{t\geq0}$ is an Itô diffusion and $(S_t)_{t\geq0}$ is its running maximum.
