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:

where $\ell_s^b(X)$ is the local time of $X$ on the surface $b$ given by:

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.