Crossings and occupation measures for a class of semimartingales

Abstract

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 $\psi$. .