Cramer's estimate for the exponential functional of a Levy process

Abstract

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{\xi_s} ds$ associated to a Levy process $(\xi_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process $\xi$, the main one being Cramer's condition, that asserts the existence of a real $\chi >0$ such that ${\Bbb E}(e^{\chi \xi_1})=1$. Then there exists $C>0$ satisfying, when $t \to +\infty$ : $$ {\Bbb P} (A_{\infty}> t) \sim C t^{-\chi} \quad . $$ This result can be applied for example to the process $\xi_t = at - S_{\alpha}(t)$ where $S_{\alpha}$ stands for the stable subordinator of index $\alpha$ ($0 < \alpha < 1$), and $a$ is a positive real (we have then $\chi=a^{1/(\alpha -1)}$).