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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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[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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