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A change-of-variable formula with local time on curves. (English) Zbl 1085.60033
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.

##### MSC:
 60H05 Stochastic integrals 60G44 Martingales with continuous parameter
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##### References:
 [4] Pedersen J.L., Peskir G. (2002). On nonlinear integral equations arising in problems of optimal stopping. Proc. Funct. Anal. VII (Dubrovnik 2001), Various Publ. Ser. No. 46, 159–175 · Zbl 1031.60030
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