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Loss Rates for Lévy Processes with Two Reflecting Barriers

by Søren Asmussen and Mats Pihlsgård
Thiele Research Reports Number 5 (May 2005)
Let $\{X_t\}$ be a Lévy process which is reflected at $0$ and $K>0$. The reflected process $\{V_t^K\}$ is constructed as $V_t^K =V_0^K +X_t +L_t^0 -L_t^K$ where $\{L_t^0\}$ and $\{L_t^K\}$ are the local times at $0$ and $K$, respectively. We consider the loss rate $\ell^K$, defined by $\ell^K = \mathbb{E}_{{\pi}_K} L_1^K$, where $\mathbb{E}_{{\pi}_K}$ is the expectation under the stationary measure ${\pi}_K$. The main result of the paper is the identification of $\ell^K$ in terms of ${\pi}_K$ and the characteristic triplet of $\{X_t\}$. We also derive asymptotics of $\ell^K$ as $K\to\infty$ when $\mathbb{E} X_1 <0$ and the Lévy measure of $\{X_t\}$ is light-tailed.
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This publication also serves as Research Reports number 457