Hölder regularity for backward stochastic Volterra integral equations and applications to numerical schemes

Abstract

We prove a Hölder-type regularity estimate for the martingale integrand of a backward stochastic Volterra integral equation (BSVIE). The estimate is formulated in $L^p(Ω)$ after averaging in $L^2$ over the first time variable, and gives an averaged Hölder estimate of order $1/2$ in the second time variable. Our approach is based on the approximation of the BSVIE by a system of BSDEs. For this system, we establish a uniform regularity estimate for the martingale component using Malliavin calculus, and then pass to the limit to obtain the result for the BSVIE. We allow for a general Malliavin differentiable free term and generator. In particular, neither is assumed to come from a forward stochastic differential equation or to have a specific functional form. We also propose an explicit Euler scheme for the approximating BSDE system and show that the regularity estimate yields a convergence rate for the resulting discrete approximation of the BSVIE.