Statistical inference for exponential functionals of L\'evy processes

In this paper, we consider the exponential functional $A_{\infty}=\int_0^\infty e^{-\xi_s}ds$ of a L{\'e}vy process $\xi_s$ and aim to estimate the characteristics of $\xi_{s}$ from the distribution of $A_{\infty}$. We present a new approach, which allows to statistically infer on the L{\'e}vy triplet of $\xi_{t}$, and study the theoretical properties of the proposed estimators. The suggested algorithms are illustrated with numerical simulations.