Chebyshev wavelet-based method for solving various stochastic optimal control problems and its application in finance

Document Type : Research Article


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


In this paper, a computational method based on parameterizing state and control variables is presented for solving Stochastic Optimal Control (SOC) problems. By using Chebyshev wavelets with unknown coefficients, state and control variables are parameterized, and then a stochastic optimal control problem is converted to a stochastic optimization problem. The expected cost functional of the resulting stochastic optimization problem is approximated by sample average approximation thereby the problem can be solved by optimization methods more easily. For facilitating and guar-anteeing convergence of the presented method, a new theorem is proved. Finally, the proposed method is implemented based on a newly designed algorithm for solving one of the well-known problems in mathematical fi-nance, the Merton portfolio allocation problem in finite horizon. The simu-lation results illustrate the improvement of the constructed portfolio return.


Main Subjects

[1] Aidoo, A.Y. and Wilson, M. A review of wavelets solution to stochastic heat equation with random inputs, Applied Mathematics, 6(14) (2015) 2226–2239.
[2] Aoki, M. Optimization of stochastic systems. Topics in discrete-time systems, Mathematics in Science and Engineering, Vol. 32. Academic Press, New York-London, 1967.
[3] Ayache, A. and Taqqu, M.S. Rate optimality of wavelet series approx-imations of fractional Brownian motion, J. Fourier Anal. Appl. 9(5), (2003) 451–471.
[4] Azzato, J.D. and Krawczyk, J. An improved MATLAB package for ap-proximating the solution to a continuous-time stochastic optimal control problem, Working paper of the Victoria University of Wellington, 2006.
[5] Cherukuri, A. Sample average approximation of conditional value-at-risk based variational inequalities, Optim. Lett. (2023) 1–26.
[6] Cinquegrana, D., Zollo, A.L., Montesarchio, M. and Bucchignani, E. A Metamodel-Based Optimization of Physical Parameters of High Resolu-tion NWP ICON-LAM over Southern Italy. Atmosphere 14(5), (2023) 788.
[7] Guariglia, E. and Guido, R.C. Chebyshev wavelet analysis, J. Funct. Spaces 2022, Art. ID 5542054, 17 pp.
[8] Hannah, L.A. Stochastic optimization, International Encyclopedia of the Social & Behavioral Sciences, 2 (2015) 473–481.
[9] Huschto, T. and Sager, S. Solving stochastic optimal control problems by a Wiener chaos approach, Vietnam J. Math. 42(1), (2014) 83–113.
[10] Jadamba, B., Khan, A.A., Migórski, S. and Sama, M. eds. Deterministic and stochastic optimal control and inverse problems, CRC Press, Boca Raton, FL, 2022.
[11] Kafash, B. and Nadizadeh, A. Solution of stochastic optimal control problems and financial applications, J. Math. Ext. 11 (2017) 27–44.
[12] Kappen, H.J. Stochastic optimal control theory, ICML, Helsinki, Rad-bound University, Nijmegen, Netherlands, 2008.
[13] Kloeden, P.E. and Platen, E. The numerical solution of stochastic dif-ferential equations, 3rd edn. Springer-Verlag Berlin Heidelberg, 2011.
[14] Korn, R. and Korn, E. Option pricing and portfolio optimization: Mod-ern methods of financial mathematics, Vol. 31. American Mathematical Soc., 2001.
[15] Kraft, H., Meyer-Wehmann, A. and Seifried, F.T.. Holger, K., Dynamic asset allocation with relative wealth concerns in incomplete markets, J. Econom. Dynam. Control 113 (2020), 103857, 20 pp.
[16] Krawczyk, J.B. A Markovian approximated solution to a portfolio man-agement problem,ITEM. Inf. Technol. Econ. Manag. No. 1 (2001), Paper 2, 33 pp.
[17] Kushner, H.J. and Dupuis, P. Numerical methods for stochastic control problems in continuous time, Second edition. Applications of Mathe-matics (New York), 24. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2001.
[18] Lan, G. and Zhou, Z. Dynamic stochastic approximation for multi-stage stochastic optimization, Math. Program. 187(1-2), (2021), Ser. A, 487–532.
[19] Ledoux, M. A note on large deviations for wiener chaos, Séminaire de Probabilités, XXIV, 1988/89, 1–14, Lecture Notes in Math., 1426, Springer, Berlin, 1990.
[20] Lisei, H. and Soos, A.Wavelet approximation of the solutions of some stochastic differential equations, Fourth Joint Conference on Applied Mathematics. Pure Math. Appl. 15 (2004), no. 2-3, 213–223 (2005).
[21] Marti, K. Stochastic optimization methods, Springer-Verlag, Berlin, 2005.
[22] Mohammadi, F. A efficient computational method for solving stochastic Itô-Volterra integral equations, TWMS J. Appl. Eng. Math. 5(2), (2015) 286–297.
[23] Mohammadi, F. and Hosseini, M. A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst. 348(8), (2011) 1787–1796.
[24] Odibat, Z., Erturk, V.S., Kumar, P., Ben Makhlouf, A. and Govindaraj, V., An implementation of the generalized differential transform scheme for simulating impulsive fractional differential equations, Hindawi Math-ematical Problems in Engineering, 2022, Article ID 8280203, (2022) 11 pages.
[25] Øksendal, B. Stochastic differential equations. An introduction with ap-plications,Sixth edition. Universitext. Springer-Verlag, Berlin, 2003.
[26] Pargaei, M. and Kumar, B.V. A 3D Haar wavelet method for a coupled degenerate system of parabolic equations with nonlinear source coupled with non-linear ODEs, Appl. Numer. Math. 185 (2023), 141–164.
[27] Petkovi, N. and and Božinović, M. The application of the dynamic pro-gramming method in investment optimization,Megatrend revija, 13(3) (2016) 171–182.
[28] Pham, H. Continuous-time stochastic control and optimization with fi-nancial applications, Stochastic Modelling and Applied Probability, 61. Springer-Verlag, Berlin, 2009.
[29] Rafiei, Z., Kafash, B. and Karbassi S.M. A new approach based on using Chebyshev wavelets for solving various optimal control problems, Com-put. Appl. Math. 37 (2018), S144–S157.
[30] Razzaghi, M. and Yousefi, S. The Legendre wavelets operational matrix of integration, Internat. J. Systems Sci. 32(4), (2001) 495–502.
[31] Simpkins, A. and Todorov, E. Practical numerical methods for stochastic optimal control of biological systems in continuous time and space, In 2009 IEEE Symposium on Adaptive Dynamic Programming and Rein-forcement Learning, pp. 212–218. IEEE, (2009).
[32] Sohrabi, S. Comparison Chebyshev wavelets method with BPFs method for solving Abel’s integral equation, Ain Shams Eng. J. 2(3-4), (2011) 249–254.
[33] Tourin, A. and Yan, R. Dynamic pairs trading using the stochastic control approach, J. Econom. Dynam. Control 37(10) (2013) 1972–1981.
[34] Yang, R., Li, W. and Liu, Y. A novel response surface method for struc-tural reliability, AIP Advances 12, (2022) 015205.
[35] Yang, Y., Heydari, M., Avazzadeh, Z. and Atangana, A., Chebyshev wavelets operational matrices for solving nonlinear variable-order frac-tional integral equations, Adv. Difference Equ. 2020, Paper No. 611, 24 pp.