We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in spaces. The main tools are new a priori estimates on Sobolev–Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.
The work of Eduardo Abi Jaber was supported by grants from Région Ile-de-France.
Christa Cuchiero gratefully acknowledges financial support by the Vienna Science and Technology Fund (WWTF) under grant MA16-021 and the Austrian Science Fund (FWF) under grant Y1235 of the START-program.
The research of Sergio Pulido benefited from the support of the Chair Markets in Transition (Fédération Bancaire Française) and the project ANR 11-LABX-0019. Sergio Pulido acknowledges support by the Europlace Institute of Finance (EIF) and the Labex Louis Bachelier, research project: “The impact of information on financial markets”.
"A weak solution theory for stochastic Volterra equations of convolution type." Ann. Appl. Probab. 31 (6) 2924 - 2952, December 2021. https://doi.org/10.1214/21-AAP1667