A weak solution theory for stochastic Volterra equations of convolution type

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing non-Lipschitz coefficients singular kernels. Our approach relies on convergence in Lp spaces. The main tools are new a priori estimates Sobolev–Slobodeckij norms of the solution, as well novel martingale problem that is equivalent to original equation. This leads generic approximation theorems spirit classical theory. also prove uniqueness path solutions additional hypotheses. To illustrate applicability our results, we consider scaling limits nonlinear Hawkes processes approximations Volterra by Markovian semimartingales.