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Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications

Wenbo Wang, Guangyan Jia

TL;DR

The paper addresses the existence and comparison theory for one-dimensional quadratic BSDEs with generators singular in $y$ and potentially unbounded terminal conditions, extending prior results via a domination method and Itô-Krylov techniques. It establishes a comprehensive set of results: existence of $L^{p}$ solutions, a comparison theorem under convexity, stability for perturbations, a nonlinear Feynman-Kac representation, and uniqueness of viscosity solutions for the associated singular PDEs. The work then leverages these theoretical developments to finance, showing how robust control criteria relate to stochastic differential utility and how g-expectation-based certainty equivalents can be characterized by singular quadratic BSDEs, with the quadratic coefficient encoding ambiguity aversion and the inverse of $\psi$ capturing absolute risk aversion. Collectively, the results broaden the applicability of BSDEs with singular and quadratic generators to complex risk-sensitive models in finance and physics, and provide tools for analyzing high-impact problems under model uncertainty. Future work points to high-dimensional extensions and further relaxations of the convexity and growth conditions on the generators.

Abstract

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators singular in $ y $. First, we establish the existence of solutions and a comparison theorem, thereby extending results in the literature. Additionally, we analyze the stability property and the Feynman-Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semilinear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and certainty equivalent based on $g$-expectation. In these applications, the coefficient of the quadratic term in the generator captures the level of ambiguity aversion and the coefficient of absolute risk aversion, respectively.

Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications

TL;DR

The paper addresses the existence and comparison theory for one-dimensional quadratic BSDEs with generators singular in and potentially unbounded terminal conditions, extending prior results via a domination method and Itô-Krylov techniques. It establishes a comprehensive set of results: existence of solutions, a comparison theorem under convexity, stability for perturbations, a nonlinear Feynman-Kac representation, and uniqueness of viscosity solutions for the associated singular PDEs. The work then leverages these theoretical developments to finance, showing how robust control criteria relate to stochastic differential utility and how g-expectation-based certainty equivalents can be characterized by singular quadratic BSDEs, with the quadratic coefficient encoding ambiguity aversion and the inverse of capturing absolute risk aversion. Collectively, the results broaden the applicability of BSDEs with singular and quadratic generators to complex risk-sensitive models in finance and physics, and provide tools for analyzing high-impact problems under model uncertainty. Future work points to high-dimensional extensions and further relaxations of the convexity and growth conditions on the generators.

Abstract

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators singular in . First, we establish the existence of solutions and a comparison theorem, thereby extending results in the literature. Additionally, we analyze the stability property and the Feynman-Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semilinear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and certainty equivalent based on -expectation. In these applications, the coefficient of the quadratic term in the generator captures the level of ambiguity aversion and the coefficient of absolute risk aversion, respectively.

Paper Structure

This paper contains 10 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notations and Existence Results
  3. Comparison Theorem
  4. Stability Result and Application to PDEs
  5. Stability Result
  6. Application to Singular Quadratic PDEs
  7. Applications in Finance
  8. Linking Robust Control to Stochastical Differential Utility
  9. Certainty Equivalent Based on g-expectation
  10. Discussion

Key Result

Theorem 2.1

Assume BSDE eq1 satisfies that $\xi >0$, $f$ is continuous in $(y,\,z)$ for each fixed $(s,\,\omega)$, and for any $(s,\,\omega,\,y,\, z) \in [0,\, T]\times \Omega \times (0,\, +\infty) \times \mathbb{R}^{1 \times d},$ where $\delta \ge 0$ is a constant, $a,\,b$, and $\gamma$ are nonnegative and progressively measurable processes with $a > 0 \text{ or } b > 0$. Then BSDE eq1 admits a solution $(

Theorems & Definitions (27)

  • Theorem 2.1
  • Lemma 2.1
  • proof : Proof of Theorem 1.
  • Remark 2.1
  • Theorem 3.1
  • proof : Proof.
  • Corollary 3.1
  • Remark 3.1
  • Corollary 3.2
  • proof : Proof.
  • ...and 17 more