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

Abstract

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{ξ_s} ds$ associated to a Levy process $(ξ_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process $ξ$, the main one being Cramer&#39;s condition, that asserts the existence of a real $χ>0$ such that ${\Bbb E}(e^{χξ_1})=1$. Then there exists $C>0$ satisfying, when $t \to +\infty$ : $$ {\Bbb P} (A_{\infty}> t) \sim C t^{-χ} \quad . $$ This result can be applied for example to the process $ξ_t = at - S_α(t)$ where $S_α$ stands for the stable subordinator of index $α$ ($0 < α< 1$), and $a$ is a positive real (we have then $χ=a^{1/(α-1)}$).