Large deviations for light-tailed Lévy bridges on short time scales

Abstract

Let $L=(L(t){)}_{t\ge 0}$ be a multivariate Lévy process with Lévy measure $\hat{I}½(dy)=\exp (-f(|y|))dy$ for a smoothly regularly varying function $f$ of index $\hat{I}\pm >1$. The process $L$ is renormalized as $X^{\epsilon }(t)=\epsilon L(r_{\epsilon }t)$, $t\in [0,T]$, for a scaling parameter $r_{\epsilon }=o({\epsilon }^{-1})$, as $\epsilon \to 0$. We study the behavior of the bridge $Y^{\epsilon ,x}$ of the renormalized process $X^{\epsilon }$ conditioned on the event $X^{\epsilon }(T)=x$ for a given end point $x\ne 0$ and end time $T>0$ in the regime of small $\epsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\epsilon ,x}$ with a specific speed function $S(\epsilon )$ and an entropy-type rate function $I_x$ on the Skorokhod space in the limit $\epsilon \to 0+$. We show that the asymptotic energy minimizing path of $Y^{\epsilon ,x}$ is the linear parametrization of the straight line between $0$ and $x$, while all paths leaving this set are exponentially negligible. We also infer a LDP for the asymptotic number of jumps and establish asymptotic normality of the jump increments of $Y^{\epsilon ,x}$. Since on these short time scales $r_{\epsilon }=o({\epsilon }^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\epsilon }(t)$, $t\in [0,T]$,for which we solve a specific nonlinear functional equation.