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 $({\xi }_t{)}_{t\ge 0}$. 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 $\mathbb{E}(e^{\chi {\xi }_1})=1$. Then there exists $C>0$ satisfying, when $t\to +\infty$ : $$\mathbb{P}(A_{\infty }>t)\sim C{t}^{-\chi }.$$ 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 $\chi =a^{1/(\alpha -1)}$).

12 pages