# A change-of-variable formula with local time on curves

Peskir, Goran (2005) A change-of-variable formula with local time on curves. Journal of Theoretical Probability, 18 (3). pp. 499-535. ISSN 1572-9230

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 C1,2 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.