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Stochastic Volterra integral equations driven by $ G $-Brownian motion

Bingru Zhao, Renxing Li, Mingshang Hu

TL;DR

The paper studies stochastic Volterra integral equations driven by $G$-Brownian motion ($G$-SVIE), establishing existence, uniqueness, and two notions of continuity for solutions under mild growth and continuity assumptions. It introduces a contraction-mapping framework in the $G$-expectation setting and proves mean-square and pathwise continuity, along with a priori stability estimates. A central contribution is a comparison theorem for $G$-SVIEs with separable diffusion coefficients, achieved via a new quasilinearization technique and a two-step approximation that relaxes classical differentiability requirements of coefficients. These results advance robust stochastic analysis under model uncertainty and memory effects, with a concrete linear example illustrating the applicability of the methods.

Abstract

In this paper, we study the stochastic Volterra integral equation driven by $G$-Brownian motion ($G$-SVIE). The existence, uniqueness and two types of continuity of the solution to $G$-SVIE are obtained. Moreover, combining a new quasilinearization technique with the two-step approximation method, we establish the corresponding comparison theorem for a class of $G$-SVIEs. In particular, by means of this method, the classical assumptions on partial derivatives of the coefficients are unnecessary.

Stochastic Volterra integral equations driven by $ G $-Brownian motion

TL;DR

The paper studies stochastic Volterra integral equations driven by -Brownian motion (-SVIE), establishing existence, uniqueness, and two notions of continuity for solutions under mild growth and continuity assumptions. It introduces a contraction-mapping framework in the -expectation setting and proves mean-square and pathwise continuity, along with a priori stability estimates. A central contribution is a comparison theorem for -SVIEs with separable diffusion coefficients, achieved via a new quasilinearization technique and a two-step approximation that relaxes classical differentiability requirements of coefficients. These results advance robust stochastic analysis under model uncertainty and memory effects, with a concrete linear example illustrating the applicability of the methods.

Abstract

In this paper, we study the stochastic Volterra integral equation driven by -Brownian motion (-SVIE). The existence, uniqueness and two types of continuity of the solution to -SVIE are obtained. Moreover, combining a new quasilinearization technique with the two-step approximation method, we establish the corresponding comparison theorem for a class of -SVIEs. In particular, by means of this method, the classical assumptions on partial derivatives of the coefficients are unnecessary.
Paper Structure (4 sections, 13 theorems, 121 equations)

This paper contains 4 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and uniqueness of the solution of G-SVIE
  4. Comparison theorem of $G$-SVIEs

Key Result

Theorem 2.1

Let $(\Omega ,L_{G}^{1}(\Omega),\hat{\mathbb{E}})$ be a $G$-expectation space. Then there exists a weakly compact set of probability measures $\mathcal{P}$ on $(\Omega,\mathcal{F})$ such that

Theorems & Definitions (25)

  • Theorem 2.1: hu2009representationdenis2011function
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: hu2014backward
  • Definition 2.5
  • Theorem 2.6: denis2011function
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 15 more