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 $\chi=a^{ 1/(\alpha -1)}$).
Categories: math.PR