Hybrid direct fuzzy quantum control design for a class of quantum stochastic systems and its application in pairs trading strategy

Document Type : Research Paper


Department of Mathematics and Computer Sciences, Lorestan University, Lorestan, Iran



In this paper, a new hybrid direct fuzzy quantum control is designed for a class of quantum stochastic systems where the dynamics of the state variable are prescribed via a Quantum Stochastic Differential Equation (QSDE) with respect to a quantum Brownian motion on a quantum probability space. The presented control is comprised of two parts, an adaptive fuzzy control part that performs the main control action and a quantum-fuzzy control part that is implemented when the existence and uniqueness of the solution are not established. Thereby, the adjusted laws of the control parameters and the quantum-fuzzy rules are designed via the Lyapunov-based technique such that the stability of the system is guaranteed. One theorem for facilitating the Fuzzy controller design algorithm is presented and proved. The proposed control method enhances the applicability of the quantum stochastic control theory for many practical control problems such as portfolio management. Therefore, theoretical results are illustrated by simulating the pairs trading problem. According to simulation results, the performance of the pairs trading strategy is improved as an increasing return portfolio that is controlled by the proposed method.


[1] W.J. Chang, C.C. Ku, and P.H. Huang, Fuzzy control of uncertain stochastic time-delay Takagi-Sugeno fuzzy models for achieving passivity, Fuzzy Sets Syst. 161 (2010), no. 15, 2012–2032.
[2] W. Chang, Y. Lin, Y. Lin, C. Pen, and M. Tsai, Saturated fuzzy controller design for interval type-2 Takagi-Sugeno fuzzy models with multiplicative noises, Processes 9 (2021), no. 5, 823.
[3] S. Chen and J. Yong, Linear quadratic optimal control problems, Appl. Math. Optim. 43 (2001), no. 11, 21–45
[4] L.A. Cotfas, A finite-dimensional quantum model for the stock market, Phys. A: Statist. Mech. Appl. 392 (2012), 371–380.
[5] T.E Dabbous, Optimal control for bilinear stochastic systems with F parameters, Dyn. Control 11 (2001), no. 1, 243—259.
[6] D. D’Alessandro, To Quantum Control and Dynamics, 1st Edition., Iowa State University Ames, U.S.A., 2008.
[7] C. Dumitrescu, P. Ciotirnae, and C. Vizitiu, Fuzzy logic for intelligent control system using soft computing applications, Sensors 21 (2021), no. 8, 1-33.
[8] B.P. Duncan, Walk Around Some Problems in Identification and Stochastic Adaptive Control with Applications to Finance, Research supported in part by NSF Grant DMS 0204669, 2004, pp. 1–26.
[9] T.E. Duncan, A direct method for solving stochastic control problems, Commun. Inf. Syst. 12 (2012), no.1, 1–14.
[10] J. Feng, W. Zhu, and Z. Ying, Optimal control of partially observable non-linear quasi-integrable Hamiltonian systems, Sci. China Phys. Mech. Astronomy 53 (2010), no. 1, 147–154.
[11] B.F. German and M.D. Berrade, Special issue “Probability theory and stochastic modeling with applications, Mathematics 11 (2023), no. 14, 3196.
[12] C. Han, Z. He, and W.J.A. Toh, Trading via unsupervised learning, Eur. J. Oper. Res. 307 (2023), no. 2, 929–947.
[13] H.J. Kappen, Integrals symmetry breaking for optimal control theory, J. Statist. Mech.: Theory Exper. 2 (2005), no. 4, 110–131.
[14] H.J. Kappen and M. Toussaint, Stochastic Optimal Control Theory, ICML, Helsinki tutorial, TU Berlin, 2008.
[15] J. Koppe, M. Patzold, M. Beyer, W. Grecksch, and W. Paul, Infinite Dimensional and Finite Dimensional Stochastic Equations and Applications in Physics, World Scientific Publishing Company, 2020.
[16] L. Minne, T. Zheng, N. Pranav, D. Ian, W. Ying, and W. Jun, Joint perception and control as inference with an object-based implementation, ICLR Conference, 2021.
[17] S. Mudchanatongsuk, J.A. Primbs, and W. Wong, Pairs Trading: A Stochastic Control Approach, IEEE Conf. Location, Seattle, WA, USA, 2008, pp. 1035–1039.
[18] F. Oreste, Trading: Using Principles of Modern Physics to Forecast the Financial Markets, 1st Edition, Wiley & Sons, Inc., Hoboken, New Jersey, 2011. [19] H. Pham, Time Stochastic Control and Optimization with Financial Applications, Springer, 2009th edition, 2009.
[20] R. Roychoudhury, R. Bhagtani, and A. Daftari, Trading using clustering and deep reinforcement learning, Available at SSRN: https://ssrn.com/abstract=4504599 or http://dx.doi.org/10.2139/ssrn.4504599
[21] A.A. Shardin and R. Wunderlich, Observable stochastic optimal control for an energy storage, Int. J. Probab. Stoch. Proces. 89 (2017), no. 1, 280–310.
[22] M. Siddiqui, M. Eddahbi, and O. Kebiri, Solutions of stochastic differential equations with jumps and measurable drifts, Mathematics 11 (2023), no. 17, 1–14.
[23] G. Wang, J. Xiong, and S. Zhang, Observable stochastic optimal control, Int. J. Numer. Anal. Model. 13 (2016), no. 3, 493–512.
[24] X. Wei, C. You, and X. Liang, option pricing formulas for fuzzy financial market based on the exponential Ornstein-Uhlenbeck model, Iran. J. Fuzzy Syst. 20 (2023), no. 4, 81–95.
[25] S. Yaghobipour and M. Yarahmadi, Control design for a class of quantum stochastic systems with financial applications, Phys. A: Statist. Mech. Appl. 512 (2018), 507-–522.
[26] S. Yaghobipour and M. Yarahmadi, Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance, Comput. Math. Appl. 79 (2019), no. 10, 2832–2845.
[27] M. Yarahmadi and S. Chegini, Design of quantum intelligent robust controller via time variant break frequency bandwidth sliding surface, Modares Mech. Engin. 17 (2017), no. 1, 305–310.
[28] B. Yongacoglu, G. Arslan, and R. Yuksel, Reinforcement learning for decentralized stochastic control, IEEE 58th Conf. Decis. Control (CDC), Nice, France, 2021, pp. 5556–5561.
[29] Q. Zhou, L. Hongyi, W. Chengwei, W. Lijie, and K. Choon, Fuzzy control of nonlinear systems with unmodeled dynamics and input saturation using small-gain approach, IEEE Trans. Syst. 47 (2017), no. 8, 100–124.

Articles in Press, Corrected Proof
Available Online from 02 February 2024
  • Receive Date: 11 September 2023
  • Accept Date: 22 December 2023