On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the $L^p$ setting, $p>0$, locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated $K-$integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the $L^p$ norm.