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Nondifferentiable functions of one-dimensional semimartingales. (English) Zbl 1193.60072

Let \(X\) be a real-valued aemimartingale and \(f:\mathbb{R}_+\times\mathbb{R}\to \mathbb{R}\). if \(f\) is twice continuously differentiable, it follows from Itô’s lemma that \[ f(t, X_t)= \int^t_0 D_x f(s, X_{s-})\,dX_s+ V_t,\tag{\(*\)} \] where \(V_t\) is a finite variation process. A real-valued process \(X\) is called a Dirichiet process if it has a decomposition \(X,Y+ V\), where \(Y\) is a semimartingale and \( \)is a càdlàg adapted zero continuous quadratic variation (z.c.q.v.) process, i.e., the quadratic variation \([V]\) of \(V\) exists, and \([V]^c= 0\). Let \({\mathcal D}_0\) denote the set of \(f: \mathbb{R}_+\times \mathbb{R}\to\mathbb{R}\) such that
(i) \(f(t,x)\) is locally Lipschitz continuuus in \(x\) and càdlàg in \(t\);
(ii) for every \(K_0< K_1\in \mathbb{R}\) and \(T\in \mathbb{R}_+\), \[ \int^{K_1}_{K_0} \int^T_0 |d_t f(t,x)|\,dx< \infty. \] If, furthermore, the left and right derivatives of \(f(t,x)\) with respect to \(x\) exist everywhere, this is written as \(f\in{\mathcal D}\). If \(f\in{\mathcal D}_0\), \(D_x f(t,x)\) is used to denote \(\limsup_{n\to 0}(f(t, x+ h)- f(t, x))/h\) which is locally bounded for any \(f\in{\mathcal D}_0\). Then the following theorem holds: Let \(X\) be a semimartingale and \(f\in{\mathcal D}\). Then \((*)\) holds where \(V\) has z.c.q.v. In particular, \(f(t,X_t)\) is a Dirichlet process. This result is a consequence of a more general decomposition result which generalizes the just mentioned theorem to arbitrary Dirichlet processes.

MSC:

60H05 Stochastic integrals
60G48 Generalizations of martingales
60G44 Martingales with continuous parameter
60G20 Generalized stochastic processes
60J60 Diffusion processes
60H20 Stochastic integral equations
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