INVERTIBILITY OF FUNCTIONALS OF THE POISSON PROCESS AND APPLICATIONS

We show that solving SDEs with constant volatility on the Wiener space is the analog of constructing Hawkes-like processes, that is, self-excited point process, on the Poisson space. Actually, both problems are linked to the invertibility of some transformations on the sample paths, which respect absolute continuity: adding an adapted drift for the Wiener space, making a random time change for the Poisson space. Following previous investigations by Üstünel on the Wiener space, we establish an entropic criterion on the Poisson space, which ensures the invertibility of such a transformation. As a consequence of this criterion, we improve the variational representation of the entropy with respect to the Poisson process distribution. Pursuing the Wiener–Poisson analogy so established, we define several notions of generalized Hawkes processes as weak or strong solutions of some fixed-point equations and show a Yamada–Watanabe like theorem for these new equations. As a consequence, we find another construction of the classical (even nonlinear) Hawkes processes without the recourse to a Poisson measure.