Noise induced oscillation in solutions of stochastic delay differential equations

Dynamic Systems and Applications

Volumn 14, Issue 2, 2005, Pages 175-196

  Appleby, John A D ^a   Buckwar, Evelyn ^b  

^a DUBLIN CITY UNIVERSITY   (Ireland)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23844489860     PISSN: 10562176     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. J. A. D. Appleby and C. Kelly. Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay. Fund. Differ. Eqn., 11(3-4):235-265, 2004.
2. J. A. D. Appleby and C. Kelly. Oscillation and non-oscillation in solutions of nonlinear stochastic delay differential equations. Electron. Comm. Probab., 9:106-118, 2004.
3. E. Beretta, V. B. Kolmanovskii, and L. Shaikhet. Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simulation, 45(3-4):269-277, 1998.
4. G. A. Bocharov and F. A. Rihan. Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math., 125(1-2):183-199, 2000.
5. Z. Brzeźniak and T. Zastawniak. Basic Stochastic Processes. A Course through Exercises. Springer, 1999.
6. M-H. Chang and R. K. Youree. The European option with hereditary price structures: Basic theory. Appl. Math. Comp., 102:279-296, 1999.
7. R.D. Driver, D.W. Sasser, and M.L. Slater. The equation x(t) - ax(t) + bx(t - τ) with small delay. American Mathematical Monthly, 80:990-995, 1973.
8. L.H. Erbe, Qingkai Kong, and B.C. Zhang. Oscillation Theory for Functional Differential Equations. Marcel Dekker Inc., New York, 1994.
9. K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Popoulation Dynamics. Kluwer Academic Publishers, 1992.
10. A. A. Gushchin and U. Küchler. On oscillations of the geometric Brownian motion with time delayed drift. Statist. Probab. Lett., 70(1): 19-24, 2004.
11. I. Györi. Invariant cones of positive initial functions for delay differential equations. Appl. Anal., 35(1-4):21-41, 1990.
12. I. Györi and G. Ladas. Oscillation Theory of Delay Differential Equations: with Applications. Clarendon Press, Oxford, 1991.
13. D. G. Hobson and L. C. G. Rogers. Complete models with stochastic volatility. Math. Finance, 8(1):27-48, 1998.
14. K. Itô and M. Nisio. On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ., 4:1-75, 1964.
15. C. Jiang, A. W. Troesch, and S. W. Shaw. Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation. Phil. Trans. R. Soc. Lond. A, 358:1761-1791, 2000.
16. I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1991.
17. T. Kato. Asymptotic behavior of solutions of the functional differential equation y′(z) = ay(λx) + by(x). In K. Schmitt, editor, Delay and Functional Differential Equations and their Applications, pages 197-217. Academic Press, 1972.
18. V. B. Kolmanovskiǐ and A. Myshkis. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht, 1999.
19. R.G. Koplatadze and T.A. Chanturiya. Oscillatory and monotone solutions of differential equations of first order with deviating argument. Differ. Uravn., 18:1463-1465, 1982.
20. Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego, 1993.
21. G. Ladas. Sharp conditions for oscillations caused by delays. Appl. Anal., 9:93-98, 1979.
22. 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, 1972.
23. G. S. Ladde and V. Lakshmikantham. Random Differential Inequalities. Academic Press, New York, 1980.
24. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang. Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, 1987.
25. G.S. Ladde. Oscillations caused by retarded perturbations of first order linear ordinary differential equations. Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 63:351-359, 1977.
26. G.S. Ladde. Stability and oscillations in single-species processes with past memory. Int. J. Syst. Sci., 10:621-647, 1979.
27. H. Lisei. Conjugation of flows for stochastic and random functional differential equations. Stochastics and Dynamics, 1(2):283-298, 2001.
28. X. Mao. Stochastic Differential Equations and their Applications. Horwood Publishing Limited, Chichester, 1997.
29. L. Markus and A. Weerasinghe. Stochastic oscillators. J. Differ. Equations, 71(2):288-314, 1988.
30. C. Masoller. Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback. Physica D, 168-169:171-176, 2002.
31. S. E. A. Mohammed. Stochastic Functional Differential Equations. Pitman, Boston, Mass., 1984.
32. A. D. Myshkis. Linear homogenous differential equations of first order with deviating arguments. Uspekhi Mat. Nauk, 5:160-162, 1950.
33. L. Shaikhet. Stability in probability of nonlinear stochastic hereditary systems. Dyn. Syst. Appl., 4(2):199-204, 1995.
34. W. E. Shreve. Oscillation in first order nonlinear retarded argument differential equations. Proc. Am. Math. Soc., 41:565-568, 1973.
35. V.A. Staikos and I.P. Stavroulakis. Bounded oscillations under the effect of retardations for differential equations of arbitrary order. Proc. Roy. Soc. Edin., 77:129-136, 1977.
36. I.P. Stavroulakis. Oscillation criteria for delay, difference and functional equations. Funct. Differ. Equ., 11(1-2):163-183, 2004.