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

Abstract

Let $L = (L(t))_{t\geq 0}$ be a multivariate Lévy process with Lévy measure $ν(dy) = \exp(-f(|y|)) dy$ for a smoothly regularly varying function $f$ of index $α>1$. The process $L$ is renormalized as $X^\varepsilon(t) = \varepsilon L(r_\varepsilon t)$, $t\in [0, T]$, for a scaling parameter $r_\varepsilon= o(\varepsilon^{-1})$, as $\varepsilon \to 0$. We study the behavior of the bridge $Y^{\varepsilon, x}$ of the renormalized process $X^\varepsilon$ conditioned on the event $X^\varepsilon(T) = x$ for a given end point $x\neq 0$ and end time $T>0$ in the regime of small $\varepsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{\varepsilon, x}$ with a specific speed function $S(\varepsilon)$ and an entropy-type rate function $I_{x}$ on the Skorokhod space in the limit $\varepsilon \rightarrow 0+$. We show that the asymptotic energy minimizing path of $Y^{\varepsilon, x}$ 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 $Y^{\varepsilon, x}$. Since on these short time scales $r_\varepsilon = o(\varepsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\varepsilon}(t)$, $t\in [0, T]$,for which we solve a specific nonlinear functional equation.