AUTHORS: Alexander Astashov, Irina Astashova, Aleksey Shamaev

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ABSTRACT: We study a second order linear differential equation with low-degree polynomial coefficients arising while studying the Bellman equation for the investment portfolio control problem. Our purpose is to determine whether there exists a non-trivial solution vanishing at infinity. We prove an existence criterion for such solutions according to the signs of the coefficients. By the way, the same methods produce an existence criterion for non-trivial bounded solutions. Instead of a verbose formulation, a united criterion is presented in a table form admitting simple computer realization.

KEYWORDS: Linear differential equations, polynomial coefficients, solutions vanishing at infinity, investment portfolio control problem

REFERENCES:

[1] Øksendal B., Stochastic Differential Equations, New-York, Springer-Verlag Heidelberg, 2000.

[2] Bielecki T., Pliska S., Risk-sensitive dynamic asset management, J. Appl. Math. and Optimiz. Math. Nachr. 1999. 37. 337–360.

[3] Bielecki T., Pliska S., Sherris M., Risk sensitive asset allocation, J. Econ. Dynamics and Contr., 2000. 24. 1145–1177.

[4] Pyatnitskii A. L., Shamaev A. S., On the asymptotic behavior of eigenvalues and eigenfunctions of non-self-adjoint elliptic operators, Journal of Mathematical Sciences, Vol. 120, No. 3, 2004.

[5] Astashova I., Qualitative properties of solutions to quasilinear ordinary differential equations (in Russian), in: I. V. Astashova (ed.), Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition, UNITY-DANA, Moscow (2012), 22–290.

[6] Astashova I. V., Asymptotic Classification of Solutions of Singular 4th-Order EmdenFowler Equations with a Constant Negative Potential, J. Math. Sci. (2018) 234:385. (Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 31, pp. 3-21, 2016.)

[7] Astashova I. V., Asymptotics of Oscillating Solutions to Equations with Power Nonlinearities, J. Math. Sci. (2018) 230:651.

[8] Astashova I., On asymptotic classification of solutions to fourth-order differential equations with singular power nonlinearity. Mathematical Modeling and Analysis, 21(4):502-521, 2016.

[9] Astashova I. V., On quasi-periodic solutions to a higher-order Emden-Fowler type differential equation. Boundary Value Problems, 174:1-8, 2014.

[10] Astashova I. V., On qualitative properties and asymptotic behavior of solutions to higher-order nonlinear differential equations. WSEAS Transactions on Mathematics, 16(5):39-47, 2017.

[11] Astashova I. V., On asymptotic classification of solutions to nonlinear regular and singular third- and fourth-order differential equations with power nonlinearity. In Differential and Difference Equations with Applications, vol. 164 of Springer Proceedings in Mathematics & Statistics, pp. 191-204. Springer International Publishing, 2016.

[12] Astashova I. V., On asymptotic equivalence of n-th order nonlinear differential equations. Tatra mountains mathematical publications, 63:3138, 2015.

[13] Astashova I. V., On asymptotic behavior of solutions to a quasi-linear second order differential equations, Functional Differential Equations, 16(1):93-115, 2009.

[14] Korchemkina T., On the behavior of solutions to second-order differential equation with general power-law nonlinearity, Memoirs on Differential Equations and Mathematical Physics. (2018) Vol. 73, pp. 101–111.

[15] Dulina K. M., On Asymptotic Behavior of Solutions to the Second-Order Emden–Fowler Type Differential Equations with Unbounded Negative Potential, Functional Differential Equations, 2016. Vol. 23. No 1–2. pp. 3–8.

[16] Bellman R., Stability theory of differential equations, McGRAW-HILL Book Company Inc., New-York–Toronto–London, 1953.

[17] Kiguradze I. T., Chanturia T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluver Academic Publishers, Dordreht-Boston-London, 1993.