Geometric BSDEs

TL;DR

This paper introduces Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs, providing theoretical foundations and applications for dynamic risk measures in finance.

Contribution

It develops the theory of GBSDEs and two-driver BSDEs, establishing existence, uniqueness, and stability results for a broad class of these equations.

Findings

Established existence and uniqueness of solutions for complex BSDEs.

Characterized auxiliary ODEs with growth involving y|ln(y)| and |z|^2/y.

Applied results to model dynamic return and risk measures.

Abstract

We introduce and develop the concepts of Geometric Backward Stochastic Differential Equations (GBSDEs, for short) and two-driver BSDEs. We demonstrate their natural suitability for modeling continuous-time dynamic return risk measures. We characterize a broad spectrum of associated, auxiliary ordinary BSDEs with drivers exhibiting growth rates involving terms of the form $y|\ln(y)|+|z|^2/y$. We establish the existence, regularity, uniqueness, and stability of solutions to this rich class of ordinary BSDEs, considering both bounded and unbounded coefficients and terminal conditions. We exploit these results to obtain corresponding results for the original two-driver BSDEs. Finally, we apply our findings within a GBSDE framework for representing the dynamics of return and star-shaped risk measures including (robust) $L^{p}$-norms, and analyze functional properties.