# Article

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Keywords:
geometric Brownian motion; delay; asymptotic decay; exponential decay
Summary:
We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and a forward-backward estimate for the square mean (exponential decay).
References:
[1] Appleby, J. A. D., Mao, X., Riedle, M.: Geometric Brownian motion with delay: Mean square characterisation. Proc. Am. Math. Soc. 137 (2009), 339-348. DOI 10.1090/S0002-9939-08-09490-2 | MR 2439458 | Zbl 1156.60045
[2] Barbălat, I.: Systèmes d'équations différentielles d'oscillations nonlinéaires. \kern-.84ptAcad. Républ. Popul. Roum., Rev. Math. Pur. Appl. 4 (1959), 267-270 French. MR 0111896 | Zbl 0090.06601
[3] El'sgol'ts, L. E., Norkin, S. B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Mathematics in Science and Engineering. Academic Press, New York (1973). DOI 10.1016/s0076-5392(08)x6170-3 | MR 0352647 | Zbl 0287.34073
[4] Erban, R., Haškovec, J., Sun, Y.: A Cucker-Smale model with noise and delay. SIAM J. Appl. Math. 76 (2016), 1535-1557. DOI 10.1137/15M1030467 | MR 3534479 | Zbl 1345.60063
[5] Fridman, E.: Tutorial on Lyapunov-based methods for time-delay systems. Eur. J. Control 20 (2014), 271-283. DOI 10.1016/j.ejcon.2014.10.001 | MR 3283869 | Zbl 1403.93158
[6] Guillouzic, S.: Fokker-Planck Approach to Stochastic Delay Differential Equations: Thesis. University of Ottawa, Ottawa (2000).
[7] Győri, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications. Oxford Mathematical Monographs. Clarendon Press, Oxford (1991). MR 1168471 | Zbl 0780.34048
[8] Hull, J. C.: Options, Futures, and other Derivatives. Prentice-Hall, Upper Saddle River (2003). Zbl 1087.91025
[9] Kolmanovskij, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Mathematics and Its Applications. Soviet Series 85. Kluwer Academic Publishers, Dordrecht (1992). DOI 10.1007/978-94-015-8084-7 | MR 1256486 | Zbl 0785.34005
[10] Kolmanovskij, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. Mathematics and Its Applications (Dordrecht) 463. Kluwer Academic Publishers, Dordrecht (1999). DOI 10.1007/978-94-017-1965-0 | MR 1680144 | Zbl 0917.34001
[11] Mao, X.: Stochastic Differential Equations and Applications. Ellis Horwood Series in Mathematics and Its Applications. Horwood Publishing, Chichester (1997). DOI 10.1533/9780857099402 | MR 1475218 | Zbl 0892.60057
[12] Mao, X.: Stability and stabilisation of stochastic differential delay equations. IET Control Theory Appl. 1 (2007), 1551-1566. DOI 10.1049/iet-cta:20070006 | MR 2352175
[13] Mohammed, S.-E. A., Scheutzow, M. K. R.: Lyapunov exponents of linear stochastic functional differential equations. II: Examples and case studies. Ann. Probab. 25 (1997), 1210-1240. DOI 10.1214/aop/1024404511 | MR 1457617 | Zbl 0885.60043
[14] ksendal, B. Ø: Stochastic Differential Equations: An Introduction with Applications. Universitext. Springer, Berlin (2003). DOI 10.1007/978-3-642-14394-6 | MR 2001996 | Zbl 1025.60026
[15] Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics 57. Springer, New York (2011). DOI 10.1007/978-1-4419-7646-8 | MR 2724792 | Zbl 1227.34001
[16] Sun, J., Chen, J.: A survey on Lyapunov-based methods for stability of linear time-delay systems. Front. Comput. Sci. 11 (2017), 555-567. DOI 10.1007/s11704-016-6120-3 | Zbl 1405.34046

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