Electronic Communications in Probability

Oscillation and Non-oscillation in Solutions of Nonlinear Stochastic Delay Differential Equations

John Appleby and Conall Kelly

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This paper studies the oscillation and nonoscillation of solutions of a nonlinear stochastic delay differential equation, where the noise perturbation depends on the current state, and the drift depends on a delayed argument. When the restoring force towards equilibrium is relatively strong, all solutions oscillate, almost surely. However, if the restoring force is superlinear, positive solutions exist with positive probability, and for suitably chosen initial conditions, the probability of positive solutions can be made arbitrarily close to unity.

Article information

Electron. Commun. Probab., Volume 9 (2004), paper no. 12, 106-118.

Accepted: 6 October 2004
First available in Project Euclid: 26 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 34K11: Oscillation theory 34K50: Stochastic functional-differential equations [See also , 60Hxx]

This work is licensed under aCreative Commons Attribution 3.0 License.


Appleby, John; Kelly, Conall. Oscillation and Non-oscillation in Solutions of Nonlinear Stochastic Delay Differential Equations. Electron. Commun. Probab. 9 (2004), paper no. 12, 106--118. doi:10.1214/ECP.v9-1115. https://projecteuclid.org/euclid.ecp/1464286692

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  • J.A.D. Appleby and E.Buckwar. Noise induced oscillation in solutions of delay differential equations. Dynam. Systems Appl. (2003), submitted.
  • J.A.D. Appleby and C. Kelly. Prevention of explosion in solutions of functional differential equations by noise perturbation. Dynam. Systems Appl. (2004), submitted.
  • J.A.D. Appleby and C.Kelly. Asymptotic and Oscillatory Properties of Linear Stochastic Delay Differential Equations with Vanishing Delay. Funct. Differ. Equ. (2004), to appear.
  • D. Dufresne. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J., 1-2 (1990), 39-79.
  • A.A. Gushchin and U. Küchler. On oscillations of the geometric Brownian motion with time delayed drift. Statist. Probab. Lett. (2003), submitted.
  • I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Second edition. Graduate Texts in Mathematics, 113. Springer, New York, 1991.
  • G. Ladas, V. Lakshmikantham, and J. S. Papadakis. Oscillations of higher-order retarded differential equations generated by the retarded argument. In K. Schmitt, editor, Delay and functional differential equations and their applications, pages 219–231. Academic Press, New York, 1972.
  • X. Mao. Stochastic differential equations and their applications. Horwood Publishing Limited, Chichester, 1997.
  • W. Shreve. Oscillation in first order nonlinear retarded argument differential equations. Proc. Amer. Math Soc., 41(2) (1973), 565–568.
  • V. Staikos and I. Stavroulakis. Bounded oscillations under the effect of retardations for differential equations of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A, 77(1-2) (1977), 129-136.