A change-of-variable formula with local time on curves (Q2576790)

Let \(X=(X_t)_{t\geq0}\) be a continuous semimartingale and let \(b:\mathbb R_+\rightarrow \mathbb R\) be a continuous function of bounded variation. Setting \(C=\{(t,x)\in \mathbb R_+\times \mathbb R\mid x<b(t)\}\) and \(D=\{(t,x)\in \mathbb R_+\times \mathbb R\mid x>b(t)\}\), 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 change-of-variable formula to \(F(t,X_t)\) with the local time of \(X\) at the curve \(b\) is proved. 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.