From hyper roughness to jumps as H→−1/2

We investigate the weak limit of the hyper-rough square-root process as the Hurst index H goes to −1/2. This limit corresponds to the fractional kernel tH−1/2 losing integrability at zero. We establish the joint convergence of the couple (X,M), where X is the hyper-rough process and M the associated martingale, to a fully correlated inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod J1 topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod M1 topology for X and in the non-Skorokhod S topology for M.