## 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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