Session S17 - Stochastic Systems: Analysis, Numerics and Applications

Tuesday, July 13, 11:40 ~ 12:15 UTC-3

Stochastic Volterra Equations

Sergio Pulido

ENSIIE, Universite Paris-Saclay, LaMME, Evry, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We obtain general weak existence and stability results for Stochastic Volterra Equations (SVEs) with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. The motivation to study SVEs comes from the literature on rough volatility models. Our approach relies on weak convergence in $L^p$ 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. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

Joint work with Eduardo Abi Jaber (Universite Paris 1 Pantheon-Sorbonne, France), Christa Cuchiero (University of Vienna, Austria) and Martin Larsson (Carnegie Mellon University, USA).

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