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Crossings and occupation measures for a class of semimartingales. (English) Zbl 0943.60019
Summary: We show that \(\frac{1} {\sqrt{\varepsilon}} \biggl\{ \int_{-\infty}^\infty f(u) k_\varepsilon N_\tau^{X_\varepsilon} (u) du- \int_0^\tau f(X_t) a_t dt \biggr\}\) converges in law (as a continuous process) to \(c_0 \int_0^\tau f(X_t) a_t dB_t\), where \(X_t= \int_0^t a_s dW_s+ \int_0^t b_s ds\), with \(W\) a standard Brownian motion, \(a\) and \(b\) regular and adapted processes, \(X_\varepsilon(t)= \int_{-\infty}^\infty (1/\varepsilon) \psi((t-u)/ \varepsilon) X_u du\), \(\psi\) a smooth kernel, \(N_t^g(u)\) the number of roots of the equation \(g(s)= u\), \(s\in (0,t]\), \(k_\varepsilon= \sqrt{\pi \varepsilon/2}/ \|\psi \|_2\), \(f\) a smooth function, \(B\) a standard Brownian motion independent of \(W\) and \(c_\psi\) a constant depending only on \(\psi\).

MSC:
60F05 Central limit and other weak theorems
60G44 Martingales with continuous parameter
60J55 Local time and additive functionals
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