A Change-of-Variable Formula with Local Time on Curves

Abstract

Let $X = (X_t)_{t \geq 0}$ be a continuous semimartingale and let $b: \mathbb{R}_+ \rightarrow \mathbb{R}$ be a continuous function of bounded variation. Setting $C = \{(t, x) \in \mathbb{R} + \times \mathbb{R} | x < b(t)\}$ and $D = \{(t,x) \in \mathbb{R}_+ \times \mathbb{R} | x > b(t)\}$ suppose that a continuous function $F: \mathbb{R}_+ \times \mathbb{R} \rightarrow \mathbb{R}$ is given such that F is C^1,2 on $\bar{C}$ and F is $C^{1,2}$ on $\bar{D}$. Then the following change-of-variable formula holds: $\eqalign{ F(t,X_t) = F(0,X_0)+\int_0^{t} {1 \over 2} (F_t(s, X_s+) + F_t(s,X_s-)) ds\cr + \int_0^t {1 \over 2} (F_x(s,X_s+) + F_x(s,X_s-))dX_s\cr + {1 \over 2} \int_0^t F_{xx} (s,X_s)I (X_s \neq b(s)) d \langle X, X \rangle_s\cr + {1 \over 2} \int_0^t (F_x(s,X_s+)-F_x(s,X_s-)) I(X_s = b(s)) d\ell_{s}^{b} (X),\cr}$ where $\ell_{s}^{b}(X)$ is the local time of X at the curve b given by $\ell_{s}^{b}(X) = \mathbb{P} - \lim_{\varepsilon \downarrow 0} {1 \over 2 \varepsilon} \int_0^s I(b(r)- \varepsilon < X_r < b(r) + \varepsilon) d \langle X, X \rangle_{r}$ and $d\ell_{s}^{b}(X)$ refers to the integration with respect to $s \mapsto \ell_{s}^{b}(X)$. A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.