Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 9 (2004) > Paper 12 open journal systems 


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

John A. D. Appleby, Dublin City University
Conall Kelly, Dublin City University, Dublin, Ireland


Abstract
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.


Full text: PDF

Pages: 106-118

Published on: October 6, 2004
Bibliography
  1. J.A.D. Appleby and E.Buckwar. Noise induced oscillation in solutions of delay differential equations. Dynam. Systems Appl. (2003), submitted. Math. Review number not available.
  2. J.A.D. Appleby and C. Kelly. Prevention of explosion in solutions of functional differential equations by noise perturbation. Dynam. Systems Appl. (2004), submitted. Math. Review number not available.
  3. 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. Math. Review number not available.
  4. D.Dufresne. The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J., 1-2 (1990), 39-79. Math. Review 92i:62195
  5. A.A. Gushchin and U. Küchler. On oscillations of the geometric Brownian motion with time delayed drift. Statist. Probab. Lett. (2003), submitted. Math. Review number not available.
  6. I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Second edition. Graduate Texts in Mathematics, 113. Springer, New York, 1991. Math. Review 92h:60127
  7. 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. Math. Review MR0372363 (51 #8579)
  8. X. Mao. Stochastic differential equations and their applications. Horwood Publishing Limited, Chichester, 1997. Math. Review MR1475218 (2000g:60099)
  9. W. Shreve. Oscillation in first order nonlinear retarded argument differential equations. Proc. Amer. Math Soc., 41(2) (1973), 565--568. Math. Review MR0372371 (51 #8587)
  10. 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. Math. Review MR0454256 (56 #12507)
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X