We present a mathematical model for credit debt repayment by a representative rational household. The household borrows a consumer loan to increase its current consumption. After taking out a consumer loan, the household chooses a debt repayment strategy that maximizes discounted consumption. The household’s objective is to maximize a utility functional comprising two components: the first represents utility derived during the debt repayment period, and the second captures the expected utility from economic preferences after the loan is fully repaid. The model is formalized as an optimal control problem featuring a non-compact control set and an endogenous time horizon. The necessary optimality conditions in the form of the Pontryagin maximum principle are investigated. We provide an analysis of the post-repayment utility functional and study the duration of the early debt repayment regime.
Keywords: mathematical modeling, optimal control, maximum principle
Received May 19, 2026
Revised June 09, 2026
Accepted June 22, 2026
Funding Agency: The work was supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the Agreement No. 075-15-2025-345
Nikolai Vsevolodovich Trusov, Cand. Sci. (Phys.-Math.), Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, 119333 Russia, e-mail: trunick.10.96@gmail.com
Alexander Alekseevich Shananin, Dr. Phys.-Math. Sci., Academician of the RAS, Prof., Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, 119333 Russia; Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia; Moscow Institute of Physics and Technology (National Research University), Dolgoprudnyi, Moscow Oblast, 141701 Russia, Peoples’ Friendship University of Russia (RUDN University), Moscow, 117198 Russia, e-mail: alexshan@yandex.ru
REFERENCES
1. Trusov N.V., Shananin A.A. Mathematical modeling of credit debt repayment strategies. Comput. Math. Math. Phys., 2026, vol. 66, no. 2, pp. 234–248. https://doi.org/10.1134/S096554252570188X
2. Trusov N.V., Shananin A.A. The impact of early loan repayment strategies on the commercial bank profitability. Lobachevskii J. Math., 2026, vol. 47, no. 2, pp. 887–903. https://doi.org/10.1134/S1995080225615243
3. Ramsey F. A mathematical theory of savings. Econ. J., 1928, vol. 38, no. 152, pp. 543–559. https://doi.org/10.2307/2224098
4. Cass D. Optimum growth in an aggregative model of capital accumulation. Rev. Econ. Stud., 1965, vol. 32, no. 3, pp. 233–240. https://doi.org/10.2307/2295827
5. Koopmans T. On the concept of optimal economic growth. In: Johansen J. (Ed.) The Econometric Approach to Development Planning. Proc. of a Study Week, 7–13 October 1963, Pontificiae Academiae Scientiarum Scripta Varia, vol. 28, Vatican City, 1965, pp. 225–300.
6. Miller B.M., Rubinovich E.Y. Impulsive control in continuous and discrete-continuous systems. NY, Springer, 2003, 447 p. https://doi.org/10.1007/978-1-4615-0095-7
7. Fisher I. The purchasing power of money, its determination and relation to credit, interest and crises (2th ed.). NY, Macmillan, 1922, 515 p. Translated to Russian under the title Pokupatel’naya sila deneg: yeyo opredeleniye i otnosheniye k kreditu, protsentu i krizisam, Moscow, Financ. Publ. House NKF USSR, 1925, 399 p.
8. Arrow K.J. Aspects of the theory of risk-bearing. Helsinki, Yrjö Jahnssonin Säätiö, 1965, 61 p.
9. Pratt J.W. Risk aversion in the small and in the large. Econometrica, 1964, vol. 32, no. 1–2, pp. 122–136.
10. Komlós J. A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar., 1967, vol. 18, iss. 1-2, pp. 217–229. https://doi.org/10.1007/BF02020976
11. Galeev E.M., Zelikin M.I., Konyagin S.V. Optimal’noe upravlenie [Optimal control]. Osmolovsky N.P., Tikhomirov V.M. (edt.). Moscow, MTsNMO, 2008, 320 p. ISBN: 978-5-94057-367-8 .
Cite this article as: N.V. Trusov, A.A. Shananin. A mathematical model of consumer loan duration under early repayment. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 3, pp. 252–266.
