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.


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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