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

TL;DR

This paper analyzes path-dependent stochastic Volterra integral equations, establishing strong well-posedness, exploring their properties, and proposing a convergent numerical scheme with explicit rate for simulation.

Contribution

It provides the first comprehensive analysis of path-dependent SVIEs, including existence, uniqueness, regularity, and a new numerical approximation method with proven convergence.

Findings

Proved strong well-posedness of path-dependent SVIEs in $L^p$ setting.

Developed an interpolated $K$-integrated Euler-Maruyama scheme.

Established convergence with explicit rate of the numerical scheme.

Abstract

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.