Maximum Principle for McKean-Vlasov Dynamic Using Lions Partial-Derivatives with Respect to Probability with Application to Finance

Hamed, Ikram; Hamed, Noureddine; Terchi, Messaouda

doi:10.22054/jmmf.2026.89775.1241

Articles in Press

Document Type : Research Article

Authors

¹ Department of Mathematics, University Mohamed Khider of Biskra, Po. Box 145, 07000 Biskra.

² Jouf University, Kingdom of Saudi Arabia KSA.

³ Laboratory of Science for Mathematics, Computer Science and Engineering Applications, Department of Mathematics, Institute of Science, University Center of Barika. Algeria

10.22054/jmmf.2026.89775.1241

Abstract

In this paper, we study optimal control problem for stochastic systems generated by general McKean-Vlasov dynamics. The coefficients of the McKean-Vlasov dynamic depend on the state of the solution process as well as of its probability law. The information available to the controller is possibly less than the whole information. Our necessary maximum principle is established by applying Girsanov's Lemma and Lions's partial-derivatives with respect to probability law. As an application to finance, the conditional random mean-variance portfolio selection problem of McKean-Vlasov type is discussed to illustrate our theoretical results, where the optimal partially observed portfolio has been derived explicitly.

Keywords

References

[1] Abada N.E.H, Hafayed M., Meherrem S. On partially observed optimal singular control of McKean-Vlasov stochastic systems: maximum principle approach Mathematical Methods in the Applied Sciences, (2022) 45(16),10363-10383.

[2] Bensoussan A, Yam S.C.P, Mean-field approach to stochastic control with partial information, Optimisation and Calculus of Variations, ESAIM: COCV 27 (2021) 89.

[3] Buckdahn R., Li J., Ma J. A stochastic maximum principle for general mean-field systems. Appl. Math. Optim. (2016) 74(3), 507--534.

[4] Chan, L. K., Karceski, J., & Lakonishok, J. On portfolio optimization: Forecasting covariances and choosing the risk model. Review of Financial Studies, 12(5), (1999) 937--974.

[5] Chang, H. Dynamic mean-variance portfolio selection with liability and stochastic interest rate. Economic Modelling, 51, (2015) 172--182.

[6] Cardaliaguet, P.: Notes on mean field games (from P.-L. Lions' lectures at Collège de France. https://www.ceremade.dauphine.fr/cardalia/ (2013),

[7] Cardaliaguet, P., Delarue, F. Lasry J-M., Lions P-L., The master equation and the convergence problem in mean field games, Annals of Mathematics Studies, Vol 201, Doi.org/10.1515/9780691193717. (2019).

[8] El Karoui, N. Peng S, Quenez MC, Backward stochastic differential equations in finance, Math Finance 7 (1) (1997), 1-71.

[9] Guenane L., Hafayed M., Meherrem S., Abbas S. On optimal solutions of general continuous-singular stochastic control problem of mean-field type. Mathematical Methods in the Applied Sciences. (2020) 43(10), 6498-6516,

[10] Hafayed, M., Abbas, S.: On near-optimal mean-field stochastic singular controls: necessary and sufficient conditions for near-optimality, J. Optim Theory Appl, 2014;160(3);778-808.

[11] Hafayed M., Meherrem S. On optimal control of mean-field stochastic systems driven by Teugels martingales via derivative with respect to measures, International journal of control, 93(5), 1053-1062 (2020).

[12] Haussmann, U.G., Suo W.: Singular optimal control I, II, SIAM J. Control Optim., 1995;33(3), 916-936, 937-959.

[13] Korichi F, Boukaf S., Hafayed M: Stochastic intervention control of mean-field Poisson jump-system with noisy observation via L-derivatives with respect to probability law. Boletim da Sociedade Paranaense de Matemática Vol 42 , (2024) 1-25.

[14] Korichi F, Hafayed M, Lions's partial derivatives with respect to probability measures for general mean-field stochastic control problem. Journal on Mathematical Modeling. Vol. 12(3) 2024, 517--532.

[15] Jeanblanc-Picqué M.: Impulse control method and exchange rate. Math Finance (1993) 3:161-177.

[16] Korn. K. Some applications of impulse control in mathematical finance, Mathematical Methods of Operations Research, 50(3) 493--518, (1999).

[17] Lions P.L. Cours au Collège de France: Théorie des jeux champs moyens. http://www.college-de-france.fr/default/EN/all/equ[1]der/audiovideo.jsp. (2013),

[18] Li. D., Zhou, X.Y. Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Applied Mathematics and Optimization, (2000) 42, 19-33.

[19] Markowitz, H. Portfolio Selection. The Journal of Finance, 7, (1952) 77-91.

[20] Miloudi H, Meherrem S, Lakhdari I.E, Hafayed M. Necessary conditions for partially observed optimal control of general McKean-Vlasov stochastic differential equations with jumps. International Journal of Control, 95(11), (2021) 3170-3181.

[21] Mundaca G., Øksendal B.: Optimal stochastic intervention control with application to the exchange rate, Journal of Mathematical Economics,1988; 29, 225-243.

[22] Pham, H. Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications, Probability, Uncertainty and Quantitative Risk (2016): 1(7), 1-26.

[23] Wang.G, Wu. Z, Xiong J. Maximum principles for forward-backward stochastic control systems with correlated state and observation noise. SIAM Journal on Control and Optimization, (2013) 51, 491-524,

[24] Wang G., Zhang C., & Zhang, W. Stochastic maximum principle for mean-field type optimal control with partial information. IEEE Transactions on Automatic Control, (2014) 59, 522-528,

[25] Wang M., Shi Q., Meng, Q. Optimal Control of Forward-Backward Stochastic Jump Diffusion Differential Systems with Observation Noises: Stochastic Maximum Principle. Asian Journal of Control. https://doi.org/10.1002/asjc.2272. (2019),

[26] Zhang L. Singular optimal controls for stochastic recursive systems under convex control constraint, Journal of Mathematical Analysis and Applications 497(2), (2021) DOI: 10.1016/j.jmaa.2020.124905.

Journal of Mathematics and Modeling in Finance

Maximum Principle for McKean-Vlasov Dynamic Using Lions Partial-Derivatives with Respect to Probability with Application to Finance

References

References

Articles in Press, Accepted Manuscript
Available Online from 19 April 2026

Maximum Principle for McKean-Vlasov Dynamic Using Lions Partial-Derivatives with Respect to Probability with Application to Finance

References

References

Articles in Press, Accepted Manuscript Available Online from 19 April 2026

Articles in Press, Accepted Manuscript
Available Online from 19 April 2026