Mean-field backward stochastic Volterra integral equations: well-posedness and related particle system
Tao Hao, Ying Hu, Jiaqiang Wen
TL;DR
The paper advances mean-field backward stochastic Volterra integral equations (mean-field BSVIEs) by establishing global well-posedness under linear and quadratic growth in $Z$, even when the generator depends on the law of $(Y,Z)$. It then analyzes particle-system approximations, proving propagation of chaos with explicit convergence rates: ${\cal Q}(N)$ in the linear-growth setting and ${\cal O}(N^{-1/(2\lambda)})$ in the quadratic-growth setting when $g$ is independent of the law of $Z$, using time-slicing, BMO martingale techniques, and Girsanov's theorem. The results extend mean-field BSVIE theory to law-dependent and diagonal-quadratic generators and provide quantitative rates for finite-$N$ approximations, with rate dependence on dimension and integrability. Together, these contributions offer new tools for analyzing large interacting systems modeled by mean-field BSVIEs in finance, economics, and stochastic control.
Abstract
This paper studies the mean-field backward stochastic Volterra integral equations (mean-field BSVIEs) and associated particle systems. We establish the existence and uniqueness of solutions to mean-field BSVIEs when the generator $g$ is of linear growth or quadratic growth with respect to $Z$, respectively. Moreover, the propagation of chaos is analyzed for the corresponding particle systems under two conditions. When $g$ is of linear growth in $Z$, the convergence rate is proven to be of order $\mathscr{Q}(N)$. When $g$ is of quadratic growth in $Z$ and is independent of the law of $Z$, we not only establish the convergence of the particle systems but also derive a convergence rate of order $\mathscr{O}(N^{-\frac{1}{2λ}})$, where $λ>1$.