Some properties of G-SVIEs
Xue Zhang, Renxing Li
TL;DR
This paper studies solvability of $G$-stochastic Volterra integral equations ($G$-SVIEs) under two coefficient regimes: time-varying Lipschitz and integral-Lipschitz, within a $G$-expectation framework. Using the Picard iteration method, it establishes existence and uniqueness of solutions $X$ to $G$-SVIEs, with $X\in M_G^{2}(0,T)$ and $\sup_{t\in[0,T]}\hat{\mathbb{E}}[|X(t)|^{2}]<\infty$, and shows $X(\cdot)-\phi(\cdot)\in \tilde{M}_G^{2}(0,T)$ is mean-square continuous under Lipschitz coefficients. For the non-Lipschitz case, driven by a concave modulus $\psi$ with $\int_{0}^{+}\frac{ds}{\psi(s)}=\infty$, a Picard-type argument yields existence and uniqueness in $\tilde{M}_G^{2}(0,T)$ and mean-square continuity. The parameterized problem is treated by assuming Lipschitz in the parameter with a bound (H5), yielding a unique $X_{\alpha}\in \tilde{M}_G^{2}(0,T)$ that is mean-square continuous in $t$ and has a quasi-continuous modification in $\alpha$; under additional regularity, $X_{\alpha}$ depends Lipschitz-continuously on $\alpha$. These results extend $G$-SVIE solvability to memory-driven models with volatility uncertainty and provide a framework for stability under perturbations.
Abstract
In this paper, we investigated the solvability of G-SVIEs under two cases: time-varying Lipschitz coefficients and integral-Lipschitz coefficients. Using the Picard iteration method, we established the existence and uniqueness of solutions to G-SVIEs under these two conditions. Additionally, we prove the continuity of the solution with respect to parameters in parameter-dependent G-SVIEs with Lipschitz coefficients.