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A change-of-variable formula with local time on surfaces. (English) Zbl 1141.60035
Donati-Martin, Catherine (ed.) et al., Séminaire de Probabilités XL. Berlin: Springer (ISBN 978-3-540-71188-9/pbk). Lecture Notes in Mathematics 1899, 69-96 (2007).
In [J. Theoret. Probab. 18, No. 3, 499–535 (2005; Zbl 1085.60033)] the author proved a Tanaka change of variable formula for functions $$f(t,X_t)$$ of a semimartingale $$(X_t)$$, where $$f$$ is $${\mathcal C}^{1,2}$$ on two domains of $${\mathbb R}_+ \times {\mathbb R}$$ that are separated by a continuous curve.
In the paper under review he proves a multidimensional extension of that result to $$f(X_t)$$ where $$X_t$$ is a $${\mathbb R}^n$$-valued semimartingale and $$f$$ is $${\mathcal C}^{i_1,\ldots ,i_n}$$ on two domains in $${\mathbb R}^n$$, $$n \geq 2$$, that are separated by a surface defined by a continuous function $$b: {\mathbb R}^{n-1} \to {\mathbb R}^n$$. The formula uses the local time of $$X_t$$ on the surface $$b$$ and the index $$i_k$$ above equals $$1$$ or $$2$$ according to whether the component $$(X^k_t)$$ is of bounded variation or not, $$k=1,\dots ,n$$.
A similar formula under weaker conditions on $$f$$ is given for $$f(t,X_t,S_t)$$, where $$(X_t)$$ is a diffusion in $${\mathbb R}$$ and $$S_t$$ is its maximum on $$[0,t]$$.
For the entire collection see [Zbl 1116.60002].

##### MSC:
 60H05 Stochastic integrals 60J55 Local time and additive functionals 60G44 Martingales with continuous parameter 60J60 Diffusion processes 60J65 Brownian motion 35R35 Free boundary problems for PDEs
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