The Annals of Probability

Crossings and occupation measures for a class of semimartingales

Gonzalo Perera and Mario Wschebor

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We show that $$\frac{1}{\sqrt{\varepsilon}}{\int_{-\infty}^{\infty} f(u)k_{\varepsilon}N_{\tau}^{X_{\varepsilon}}(u)du - \int_0^{\tau} f(X_t)a_t dt}$$ converges in law (as a continuous process) to $c_{\psi} \int_0^{\tau}f(X_t)a_t dB_t$ where $X_t = \int_0^t a_s dW_s + \int_0^t b_x ds$, 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 $g(s) = u, s \epsilon (o, t], k_{\varepsilon} = \sqrt{\pi \varepsilon /2/ \parallel \psi \parallel_2$, $f$ a smooth function, a standard Brownian motion independent of $W$ and $c_{\psi}$ constant depending only on $\psi$. .

Article information

Ann. Probab., Volume 26, Number 1 (1998), 253-266.

First available in Project Euclid: 31 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Crossings local time occupation measure semimartingale smoothing of paths


Perera, Gonzalo; Wschebor, Mario. Crossings and occupation measures for a class of semimartingales. Ann. Probab. 26 (1998), no. 1, 253--266. doi:10.1214/aop/1022855418.

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