Asymptotic results for exponential functionals of Levy processes

In this work we give a complete description to the asymptotic behaviors of exponential functionals of L\'evy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds are proved, the accurate limit constants are also given. As an application, we study the survival probabilities of continuous-state branching processes in random environment defined in He et al. (2016). Like the discrete case and branching diffusion in random environment, we classify them into five different types according to their extinction speeds.