Abstract semilinear stochastic Itó-Volterra
integrodifferential equations

Keck, David N.; McKibben, Mark A.

doi:https://doi.org/10.1155/JAMSA/2006/45253

International Journal of Stochastic Analysis

On this page

Abstract References Copyright Related Articles

Open Access

Volume 2006 | Article ID 045253 | https://doi.org/10.1155/JAMSA/2006/45253

Abstract semilinear stochastic Itó-Volterra integrodifferential equations

David N. Keck¹and Mark A. McKibben²

Received31 Oct 2005

Revised03 Mar 2006

Accepted14 Apr 2006

Published04 Jul 2006

Abstract

We consider a class of abstract semilinear stochastic Volterra integrodifferential equations in a real separable Hilbert space. The global existence and uniqueness of a mild solution, as well as a perturbation result, are established under the so-called Caratheodory growth conditions on the nonlinearities. An approximation result is then established, followed by an analogous result concerning a so-called McKean-Vlasov integrodifferential equation, and then a brief commentary on the extension of the main results to the time-dependent case. The paper ends with a discussion of some concrete examples to illustrate the abstract theory.

References

N. U. Ahmed and X. Ding, “A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space,” Stochastic Processes and Their Applications, vol. 60, no. 1, pp. 65–85, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Aizicovici and K. B. Hannsgen, “Local existence for abstract semilinear Volterra integrodifferential equations,” Journal of Integral Equations and Applications, vol. 5, no. 3, pp. 299–313, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Aizicovici and M. A. McKibben, “Semilinear Volterra integrodifferential equations with nonlocal initial conditions,” Abstract and Applied Analysis, vol. 4, no. 2, pp. 127–139, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. T. Bharucha-Reid, Random Integral Equations, vol. 96 of Mathematics in Science and Engineering, Academic Press, New York, 1972.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
R. W. Carr and K. B. Hannsgen, “A nonhomogeneous integro-differential equation in Hilbert space,” SIAM Journal on Mathematical Analysis, vol. 10, no. 5, pp. 961–984, 1979.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. W. Carr and K. B. Hannsgen, “Resolvent formulas for a Volterra equation in Hilbert space,” SIAM Journal on Mathematical Analysis, vol. 13, no. 3, pp. 459–483, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Gh. Constantin, “The uniqueness of solutions of perturbed backward stochastic differential equations,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 12–16, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Da Prato and M. Iannelli, “Linear integro-differential equations in Banach spaces,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 62, pp. 207–219, 1980.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. A. Dawson, “Critical dynamics and fluctuations for a mean-field model of cooperative behavior,” Journal of Statistical Physics, vol. 31, no. 1, pp. 29–85, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
D. A. Dawson and J. Gärtner, “Large deviations from the McKean-Vlasov limit for weakly interacting diffusions,” Stochastics, vol. 20, no. 4, pp. 247–308, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. El Boukfaoui and M. Erraoui, “Remarks on the existence and approximation for semilinear stochastic differential equations in Hilbert spaces,” Stochastic Analysis and Applications, vol. 20, no. 3, pp. 495–518, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. C. Grimmer, “Resolvent operators for integral equations in a Banach space,” Transactions of the American Mathematical Society, vol. 273, no. 1, pp. 333–349, 1982.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. B. Hannsgen and R. L. Wheeler, “Behavior of the solution of a Volterra equation as a parameter tends to infinity,” Journal of Integral Equations, vol. 7, no. 3, pp. 229–237, 1984.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. Hu and N. Lerner, “On the existence and uniqueness of solutions to stochastic equations in infinite dimension with integral-Lipschitz coefficients,” Journal of Mathematics of Kyoto University, vol. 42, no. 3, pp. 579–598, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Ichikawa, “Stability of semilinear stochastic evolution equations,” Journal of Mathematical Analysis and Applications, vol. 90, no. 1, pp. 12–44, 1982.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Kannan, “Random integrodifferential equations,” in Probabilistic Analysis and Related Topics, Vol. 1, A. R. Bharucha-Reid, Ed., pp. 87–167, Academic Press, New York, 1978.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. Kannan and A. T. Bharucha-Reid, “On a stochastic integro-differential evolution equation of Volterra type,” Journal of Integral Equations, vol. 10, no. 1–3, suppl., pp. 351–379, 1985.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
D. N. Keck and M. A. McKibben, “Abstract stochastic integrodifferential delay equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2005, no. 3, pp. 275–305, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
J. A. León and D. Nualart, “Stochastic evolution equations with random generators,” The Annals of Probability, vol. 26, no. 1, pp. 149–186, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Y. P. Lin and J. H. Liu, “Semilinear integrodifferential equations with nonlocal Cauchy problem,” Nonlinear Analysis, vol. 26, no. 5, pp. 1023–1033, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. H. Liu, “Integrodifferential equations with non-autonomous operators,” Dynamic Systems and Applications, vol. 7, no. 3, pp. 427–439, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. H. Liu and K. Ezzinbi, “Non-autonomous integrodifferential equations with non-local conditions,” Journal of Integral Equations and Applications, vol. 15, no. 1, pp. 79–93, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Yu. S. Mīshura, “Existence of solutions of abstract Volterra equations in a Banach space and its subsets,” Ukraïns' kiĭ Matematichniĭ Zhurnal, vol. 52, no. 5, pp. 648–657, 2000, English translation in Ukrainian Mathematical Journal \textbf{52} (2000), no. 5, 741–753 (2001).
View at: Google Scholar | MathSciNet
Yu. S. Mīshura, “Abstract Volterra equations with stochastic kernels,” Theory of Probability and Mathematical Statistics, no. 64, pp. 139–151, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Yu. S. Mīshura and Yu. V. Tomīlov, “The method of successive approximations for abstract Volterra equations in a Banach space,” Ukraïns' kiĭ Matematichniĭ Zhurnal, vol. 51, no. 3, pp. 376–382, 1999, English translation in Ukrainian Mathematical Journal \textbf{51} (1999), no. 3, 419–426.
View at: Google Scholar | MathSciNet
M. Nagasawa and H. Tanaka, “Diffusion with interactions and collisions between coloured particles and the propagation of chaos,” Probability Theory and Related Fields, vol. 74, no. 2, pp. 161–198, 1987.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Prüss, “Positivity and regularity of hyperbolic Volterra equations in Banach spaces,” Mathematische Annalen, vol. 279, no. 2, pp. 317–344, 1987.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
J. Prüss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkhäuser, Basel, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 David N. Keck and Mark A. McKibben. 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.

PDF Download Citation Order printed copies

Views

187

Downloads

644

Citations