A G-BSDE approach to the long-term decomposition of robust pricing kernels

TL;DR

This paper introduces a G-BSDE approach for decomposing long-term pricing kernels under volatility uncertainty, extending classical models to a G-expectation framework with new existence, uniqueness, and representation results.

Contribution

It develops a novel G-BSDE methodology for long-term pricing kernel decomposition, incorporating volatility uncertainty and linking solutions to second-order PDEs.

Findings

Decomposition of pricing kernels into four components under G-expectation.

Existence and uniqueness of solutions for quadratic G-BSDEs.

Representation of components via second-order PDEs.

Abstract

This study proposes a BSDE approach to the long-term decomposition of pricing kernels under the G-expectation framework. We establish the existence, uniqueness, and regularity of solutions to three types of quadratic G-BSDEs: finite-horizon G-BSDEs, infinite-horizon G-BSDEs, and ergodic G-BSDEs. Moreover, we explore the Feynman--Kac formula associated with these three types of quadratic G-BSDEs. Using these results, a pricing kernel is uniquely decomposed into four components: an exponential discounting component, a transitory component, a symmetric G-martingale, and a decreasing component that captures the volatility uncertainty of the G-Brownian motion. Furthermore, these components are represented through the solution to a second-order PDE. This study extends previous findings obtained under a single fixed probability framework to the G-expectation context.