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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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