A maximum principle for robust optimal control problems of quadratic BSDE

Abstract

The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations (BSDEs). The system coefficients depend on parameter 𝜃, while the generator of BSDEs exhibits quadratic growth with respect to 𝑧. To address the uncertainty present in the model, the variational inequality is derived using weak convergence techniques. Additionally, due to the generator being quadratic with respect to 𝑧, the forward adjoint equations are stochastic differential equations with unbounded coefficients, involving mean oscillation martingales. By using the reverse Hölder inequality and John−Nirenberg inequality, we demonstrate that the solutions are continuous with respect to parameter 𝜃. Moreover, the necessary and sufficient conditions for robust optimal control are established using the linearization method.

Keywords

Quadratic BSDE / Model uncertainty / Maximum principle / Robust optimal control

Acknowledgements

This paper is supported by the National Key R&D Program of China (Grant Nos. 2022YFA1006101 and 2022YFA1006102), National Natural Science Foundation of China (Grant Nos. 72171133, 12101291, and 12371445), Natural Science Foundation of Shandong Province (Grant Nos. ZR2024MA039 and ZR2022MA029), Guangdong Basic and Applied Basic Research Foundation (Grant No. 2025B1515020091), Shenzhen Fundamental Research General Program (Grant No. JCYJ20230807093309021), and Science and Technology Commission of Shanghai Municipality (Grant No. 22ZR1407600).

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