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Backward Stochastic Volterra integral equations driven by G-Brownian motion

Bingru Zhao, Mingshang Hu

TL;DR

This work extends backward stochastic Volterra integral equations to the nonlinear $G$-expectation setting, formulating $G$-BSVIEs driven by $G$-Brownian motion to capture volatility and model uncertainty. A novel backward iteration scheme constructs local Picard sequences on subintervals of $[0,T]$ and then stitches them to a global solution, with existence, uniqueness, and continuity proven via $G$-stochastic analysis and monotone convergence. A comparison theorem is established under suitable monotonicity, providing a tool for risk-sensitive stochastic control and dynamic risk measures under nonlinear expectations. The results deliver a rigorous well-posedness framework for $G$-BSVIEs, enabling applications in stochastic control under volatility ambiguity and robust utility models.

Abstract

In this paper, we study the Backward stochastic Volterra integral equation driven by G-Brownian motion (G-BSVIE). By adopting a different backward iteration method, we construct the approximating sequences on each local interval. With the help of G-stochastic analysis techniques and the monotone convergence theorem, the existence, uniqueness, and continuity of the solution over the entire interval are established. Moreover, we derive the comparison theorem.

Backward Stochastic Volterra integral equations driven by G-Brownian motion

TL;DR

This work extends backward stochastic Volterra integral equations to the nonlinear -expectation setting, formulating -BSVIEs driven by -Brownian motion to capture volatility and model uncertainty. A novel backward iteration scheme constructs local Picard sequences on subintervals of and then stitches them to a global solution, with existence, uniqueness, and continuity proven via -stochastic analysis and monotone convergence. A comparison theorem is established under suitable monotonicity, providing a tool for risk-sensitive stochastic control and dynamic risk measures under nonlinear expectations. The results deliver a rigorous well-posedness framework for -BSVIEs, enabling applications in stochastic control under volatility ambiguity and robust utility models.

Abstract

In this paper, we study the Backward stochastic Volterra integral equation driven by G-Brownian motion (G-BSVIE). By adopting a different backward iteration method, we construct the approximating sequences on each local interval. With the help of G-stochastic analysis techniques and the monotone convergence theorem, the existence, uniqueness, and continuity of the solution over the entire interval are established. Moreover, we derive the comparison theorem.
Paper Structure (8 sections, 20 theorems, 139 equations)

This paper contains 8 sections, 20 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $G$-expectation
  4. Backward Stochastic differential equations driven by $G$-Brownian motion
  5. Backward Stochastic Volterra integral equations driven by $G$-Brownian motion
  6. The $G$-BSVIEs with generators independent of Y
  7. The $G$-BSVIEs with generators depending on $Y$
  8. Comparison of Backward Stochastic Volterra integral equations driven by $G$-Brownian motion

Key Result

Theorem 2.1

There exists a unique weakly compact convex collection of probability measures $\mathcal{P}$ on $\left( \Omega _{T},\mathcal{B}\left( \Omega _{T}\right) \right)$ such that where $\mathcal{B}\left( \Omega _{T}\right) =\sigma \left( B_{s}:s\leq T\right)$.

Theorems & Definitions (26)

  • Theorem 2.1: Denis2011function
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: Denis2011function
  • Proposition 2.5: Hu2014aHu2022degenerate
  • Proposition 2.6: Hu2014aHu2022degenerate
  • Theorem 2.7: Hu2014aHu2022degenerate
  • Theorem 2.8: Hu2014b
  • Theorem 2.9: Soner2011song
  • Definition 3.1
  • ...and 16 more