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Mathematics
Volume 14
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10.3390/math14010003
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Optimal Control in Financial Markets for the Uncertain Volatility Model

by
Grigory Belyavski
1,
Natalia Danilova
1,*,
Irina Zemlyakova
1 and
Gennady Ougolnitsky
2
1
Department of Optimization Methods and Machine Learning, Southern Federal University, 344090 Rostov-on-Don, Russia
2
Department of Applied Mathematics and Programming, Southern Federal University, 344090 Rostov-on-Don, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(1), 3; https://doi.org/10.3390/math14010003
Submission received: 27 October 2025 / Revised: 15 December 2025 / Accepted: 16 December 2025 / Published: 19 December 2025
Versions Notes

Abstract

This paper generalizes the well-known Black–Scholes model, specifically the uncertain volatility model. To calculate the fair price range of a payment obligation, Hamilton–Jacobi–Bellman equations are derived and transformed into nonlinear heat equations with boundary conditions. Theorems are proven stating that, for a certain class of payment obligations, solutions to nonlinear heat equations satisfy the linear heat equations. A computational example using real data is provided.
Keywords: Black-Scholes model; Heston model; uncertain volatility model; G-heat conductivity equation; viscosity solution Black-Scholes model; Heston model; uncertain volatility model; G-heat conductivity equation; viscosity solution

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MDPI and ACS Style

Belyavski, G.; Danilova, N.; Zemlyakova, I.; Ougolnitsky, G. Optimal Control in Financial Markets for the Uncertain Volatility Model. Mathematics 2026, 14, 3. https://doi.org/10.3390/math14010003

AMA Style

Belyavski G, Danilova N, Zemlyakova I, Ougolnitsky G. Optimal Control in Financial Markets for the Uncertain Volatility Model. Mathematics. 2026; 14(1):3. https://doi.org/10.3390/math14010003

Chicago/Turabian Style

Belyavski, Grigory, Natalia Danilova, Irina Zemlyakova, and Gennady Ougolnitsky. 2026. "Optimal Control in Financial Markets for the Uncertain Volatility Model" Mathematics 14, no. 1: 3. https://doi.org/10.3390/math14010003

APA Style

Belyavski, G., Danilova, N., Zemlyakova, I., & Ougolnitsky, G. (2026). Optimal Control in Financial Markets for the Uncertain Volatility Model. Mathematics, 14(1), 3. https://doi.org/10.3390/math14010003

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