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Quadratic Mean-Field BSDEs and Exponential Utility Maximization

Yining Ding, Kihun Nam, Jiaqiang Wen

TL;DR

The paper tackles mean-field backward SDEs with quadratic growth in the control and a mean-field term, proving existence and uniqueness for bounded terminals via a Malliavin calculus plus BMO stability approach, without relying on fixed-point arguments. It first handles separately quadratic drivers and then extends to a fully coupled quadratic regime under a small terminal condition, bridging prior results in the literature. A key contribution is the application to mean-field exponential utility maximization under aggregated market signals, generalizing Hu et al. (2005) to fully coupled MF settings. The results provide a solid theoretical foundation for MF financial models with collective effects and constraint-driven trading, with potential implications for risk management and optimal investment under mean-field interactions.

Abstract

In this paper, we study a class of real-valued mean-field backward stochastic differential equations (BSDEs) with generator of quadratic growth in the control variable and the mean-field term. Under this assumption, together with a bounded terminal condition, we establish existence and uniqueness of solutions. Our approach departs from classical fixed-point arguments and instead combines Malliavin calculus with refined BMO and stability estimates. The result bridges the gap between the quadratic BSDE results of Cheridito and Nam (2017) and Hao et al. (2025). Moreover, motivated by the structure of the mean-field exponential utility maximization problem introduced in our paper, we extend our framework to generators satisfying a weaker quadratic condition on the generator. This relaxation is designed to accommodate the additional mean-field terms that arise in our utility maximization setting and that fall outside the scope of previous quadratic assumptions. Within this more general regime, we establish existence and uniqueness of solutions under a smallness condition on the terminal random variable. We then apply this extended theory to solve a mean-field exponential utility maximization problem, thereby generalizing the classical framework of Hu et al. (2005)to a fully coupled quadratic mean-field setting.

Quadratic Mean-Field BSDEs and Exponential Utility Maximization

TL;DR

The paper tackles mean-field backward SDEs with quadratic growth in the control and a mean-field term, proving existence and uniqueness for bounded terminals via a Malliavin calculus plus BMO stability approach, without relying on fixed-point arguments. It first handles separately quadratic drivers and then extends to a fully coupled quadratic regime under a small terminal condition, bridging prior results in the literature. A key contribution is the application to mean-field exponential utility maximization under aggregated market signals, generalizing Hu et al. (2005) to fully coupled MF settings. The results provide a solid theoretical foundation for MF financial models with collective effects and constraint-driven trading, with potential implications for risk management and optimal investment under mean-field interactions.

Abstract

In this paper, we study a class of real-valued mean-field backward stochastic differential equations (BSDEs) with generator of quadratic growth in the control variable and the mean-field term. Under this assumption, together with a bounded terminal condition, we establish existence and uniqueness of solutions. Our approach departs from classical fixed-point arguments and instead combines Malliavin calculus with refined BMO and stability estimates. The result bridges the gap between the quadratic BSDE results of Cheridito and Nam (2017) and Hao et al. (2025). Moreover, motivated by the structure of the mean-field exponential utility maximization problem introduced in our paper, we extend our framework to generators satisfying a weaker quadratic condition on the generator. This relaxation is designed to accommodate the additional mean-field terms that arise in our utility maximization setting and that fall outside the scope of previous quadratic assumptions. Within this more general regime, we establish existence and uniqueness of solutions under a smallness condition on the terminal random variable. We then apply this extended theory to solve a mean-field exponential utility maximization problem, thereby generalizing the classical framework of Hu et al. (2005)to a fully coupled quadratic mean-field setting.

Paper Structure

This paper contains 12 sections, 15 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of main results
  3. Preliminaries
  4. Statement of the main theorem
  5. Terminal conditions with bounded Malliavin derivative
  6. A priori estimates and stability for BSDEs
  7. Proof: existence and uniqueness
  8. Application: Utility Maximization
  9. Financial market model setup
  10. Constraint set and aggregated market signal
  11. Existence and uniqueness for BSDEs with aggregated market signal
  12. Conclusion

Key Result

Lemma 2.1

Let $M$ be a one–dimensional continuous $\mathrm{BMO}$ martingale under $\mathbb P$. Then the Doléans–Dade exponential $\mathcal{E}(M)$ satisfies the reverse Hölder inequality: there exists $q^*>1$ such that for all $q\in(1,q^*)$ and all stopping times $\tau\le T$, where the constants $q^*$ and $C_q^*$ depend explicitly on the $\mathrm{BMO}$ norm $\|M\|_{\mathrm{BMO}(\mathbb{P})}$: and for $

Theorems & Definitions (36)

  • Lemma 2.1: Reverse Hölder Inequality (RHI)
  • Lemma 2.2
  • Definition 2.3: sliceability
  • Example 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • ...and 26 more