Approximations for adapted M-solutions of Type-II backward stochastic Volterra integral equations

TL;DR

This paper develops approximation methods for Type-II backward stochastic Volterra integral equations, including a BSDE-based convergence approach and a backward Euler--Maruyama numerical scheme with proven convergence rates.

Contribution

It introduces novel approximation techniques for adapted M-solutions of Type-II BSVIEs, including convergence analysis without differentiability assumptions.

Findings

BSDE approximation converges to the M-solution.

The numerical scheme converges with order 1/2 in L^2.

Provides L^2-regularity estimates for solutions.

Abstract

In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler--Maruyama scheme, and show that the scheme converges in the strong $L^2$-sense with the convergence speed of order $1/2$. These results hold true without any differentiability conditions for the coefficients.