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International Journal of Differential Equations
Volume 2010 (2010), Article ID 508217, 25 pages
http://dx.doi.org/10.1155/2010/508217
Research Article

On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations

Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

Received 1 November 2009; Accepted 14 January 2010

Academic Editor: Elena Braverman

Copyright © 2010 John A. D. Appleby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Beretta, V. Kolmanovskii, and L. Shaikhet, “Stability of epidemic model with time delays influenced by stochastic perturbations,” Mathematics and Computers in Simulation, vol. 45, no. 3-4, pp. 269–277, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. G. A. Bocharov and F. A. Rihan, “Numerical modelling in biosciences using delay differential equations,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 183–199, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M.-H. Chang and R. K. Youree, “The European option with hereditary price structures: basic theory,” Applied Mathematics and Computation, vol. 102, no. 2-3, pp. 279–296, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at MathSciNet
  5. D. G. Hobson and L. C. G. Rogers, “Complete models with stochastic volatility,” Mathematical Finance, vol. 8, no. 1, pp. 27–48, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Jiang, A. W. Troesch, and S. W. Shaw, “Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation,” Philosophical Transactions of The Royal Society of London. Series A, vol. 358, no. 1771, pp. 1761–1791, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. V. B. Kolmanovskiĭ and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
  8. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  9. C. Masoller, “Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback,” Physica D, vol. 168-169, pp. 171–176, 2002. View at Publisher · View at Google Scholar · View at Scopus
  10. L. Shaĭkhet, “Stability in probability of nonlinear stochastic hereditary systems,” Dynamic Systems and Applications, vol. 4, no. 2, pp. 199–204, 1995. View at Zentralblatt MATH · View at MathSciNet
  11. L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, , NY, USA, 1994.
  12. I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 1991. View at MathSciNet
  13. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1987. View at MathSciNet
  14. I. P. Stavroulakis, “Oscillation criteria for delay, difference and functional equations,” Functional Differential Equations, vol. 11, no. 1-2, pp. 163–183, 2004. View at Zentralblatt MATH · View at MathSciNet
  15. K. Gopalsamy and B. S. Lalli, “Necessary and sufficient conditions for “zero crossing” in integrodifferential equations,” The Tohoku Mathematical Journal. Second Series, vol. 43, no. 2, pp. 149–160, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. I. Györi and G. Ladas, “Positive solutions of integro-differential equations with unbounded delay,” Journal of Integral Equations and Applications, vol. 4, no. 3, pp. 377–390, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  17. X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, UK, 1997. View at MathSciNet
  18. J. A. D. Appleby and C. Kelly, “Oscillation and non-oscillation in solutions of nonlinear stochastic delay differential equations,” Electronic Communications in Probability, vol. 9, pp. 106–108, 2004. View at Zentralblatt MATH · View at MathSciNet
  19. J. A. D. Appleby and C. Kelly, “Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay,” Functional Differential Equations, vol. 11, no. 3-4, pp. 235–265, 2004. View at Zentralblatt MATH · View at MathSciNet
  20. J. A. D. Appleby and E. Buckwar, “Noise induced oscillation in solutions of stochastic delay differential equations,” Dynamic Systems and Applications, vol. 14, no. 2, pp. 175–195, 2005. View at Zentralblatt MATH · View at MathSciNet
  21. A. A. Gushchin and U. Küchler, “On oscillations of the geometric Brownian motion with time-delayed drift,” Statistics & Probability Letters, vol. 70, no. 1, pp. 19–24, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. A. D. Appleby, A. Rodkina, and C. Swords, “Fat tails and bubbles in a discrete time model of an inefficient financial market,” in Dynamic Systems and Applications, vol. 5, pp. 35–45, Dynamic, Atlanta, Ga, USA, 2008. View at MathSciNet
  23. J. A. D. Appleby and C. Swords, “Asymptotic behaviour of a nonlinear stochastic difference equation modelling an inefficient financial market,” Advanced Studies in Pure Mathematics, vol. 53, pp. 23–33, 2009, ICDEA2006.
  24. J.-P. Bouchaud and R. Cont, “A Langevin approach to stock market fluctuations and crashes,” European Physical Journal B, vol. 6, no. 4, pp. 543–550, 1998. View at Scopus
  25. M. A. Berger and V. J. Mizel, “Volterra equations with Itô integrals. I,” Journal of Integral Equations, vol. 2, no. 3, pp. 187–245, 1980. View at Zentralblatt MATH · View at MathSciNet
  26. V. A. Staikos and I. P. Stavroulakis, “Bounded oscillations under the effect of retardations for differential equations of arbitrary order,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 77, no. 1-2, pp. 129–136, 1977. View at Zentralblatt MATH · View at MathSciNet
  27. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1991. View at MathSciNet