- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Differential Equations
Volume 2010 (2010), Article ID 508217, 25 pages
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.
- 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.
- 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.
- 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.
- 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.
- D. G. Hobson and L. C. G. Rogers, “Complete models with stochastic volatility,” Mathematical Finance, vol. 8, no. 1, pp. 27–48, 1998.
- 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.
- V. B. Kolmanovskiĭ and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.
- Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993.
- C. Masoller, “Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback,” Physica D, vol. 168-169, pp. 171–176, 2002.
- L. Shaĭkhet, “Stability in probability of nonlinear stochastic hereditary systems,” Dynamic Systems and Applications, vol. 4, no. 2, pp. 199–204, 1995.
- L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, , NY, USA, 1994.
- I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 1991.
- 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.
- I. P. Stavroulakis, “Oscillation criteria for delay, difference and functional equations,” Functional Differential Equations, vol. 11, no. 1-2, pp. 163–183, 2004.
- 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.
- 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.
- X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood Publishing Limited, Chichester, UK, 1997.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.