Table of Contents
International Journal of Stochastic Analysis
Volume 2012, Article ID 236327, 24 pages
http://dx.doi.org/10.1155/2012/236327
Research Article

Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Received 4 June 2012; Accepted 6 August 2012

Academic Editor: Hari Mohan Srivástava

Copyright © 2012 Stefan Tappe. 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. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, Mass, USA, 1992. View at Publisher · View at Google Scholar
  2. C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, vol. 1905 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2007.
  3. X. Zhang, “Stochastic Volterra equations in Banach spaces and stochastic partial differential equation,” Journal of Functional Analysis, vol. 258, no. 4, pp. 1361–1425, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. Gawarecki and V. Mandrekar, Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations, Probability and Its Applications, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar
  5. C. Marinelli, C. Prévôt, and M. Röckner, “Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise,” Journal of Functional Analysis, vol. 258, no. 2, pp. 616–649, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. D. Filipović, S. Tappe, and J. Teichmann, “Jump-diffusions in Hilbert spaces: existence, stability and numerics,” Stochastics, vol. 82, no. 5, pp. 475–520, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Kotelenez, “A submartingale type inequality with applications to stochastic evolution equations,” Stochastics, vol. 8, no. 2, pp. 139–151, 1982/83. View at Publisher · View at Google Scholar
  8. P. Kotelenez, “A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations,” Stochastic Analysis and Applications, vol. 2, no. 3, pp. 245–265, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. C. Knoche, “SPDEs in infinite dimensional with Poisson noise,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 339, no. 9, pp. 647–652, 2004. View at Publisher · View at Google Scholar
  10. S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, vol. 113 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, Mass, USA, 2007. View at Publisher · View at Google Scholar
  11. S. Albeverio, V. Mandrekar, and B. Rüdiger, “Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise,” Stochastic Processes and Their Applications, vol. 119, no. 3, pp. 835–863, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. C. I. Prévôt, “Existence, uniqueness and regularity w.r.t. the initial condition of mild solutions of SPDEs driven by Poisson noise,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 13, no. 1, pp. 133–163, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 2003.
  14. G. Cao, K. He, and X. Zhang, “Successive approximations of infinite dimensional SDEs with jump,” Stochastics and Dynamics, vol. 5, no. 4, pp. 609–619, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, Mass, USA, 2005.
  16. C. Dellacherie and P. A. Meyer, Probabilités et Potentiel, Hermann, Paris, France, 1982.
  17. R. K. Getoor, “On the construction of kernels,” in Séminaire de Probabilités, IX, Lecture Notes in Mathematics 465, pp. 443–463, Springer, Berlin, Germany, 1975. View at Google Scholar · View at Zentralblatt MATH
  18. D. Filipović, S. Tappe, and J. Teichmann, “Term structure models driven by Wiener processes and Poisson measures: existence and positivity,” SIAM Journal on Financial Mathematics, vol. 1, pp. 523–554, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. D. Filipovic, S. Tappe, and J. Teichmann, “Invariant manifolds with boundary for jump diffusions,” http://arxiv.org/abs/1202.1076.
  20. 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 Publisher · View at Google Scholar
  21. V. Mandrekar and B. Rüdiger, “Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces,” Stochastics, vol. 78, no. 4, pp. 189–212, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. T. Yamada, “On the successive approximation of solutions of stochastic differential equations,” Journal of Mathematics of Kyoto University, vol. 21, no. 3, pp. 501–515, 1981. View at Google Scholar · View at Zentralblatt MATH
  23. T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,” Journal of Differential Equations, vol. 96, no. 1, pp. 152–169, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, The Netherlands, 1970.
  25. E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, UK, 1976.
  26. E. Hausenblas and J. Seidler, “Stochastic convolutions driven by martingales: maximal inequalities and exponential integrability,” Stochastic Analysis and Applications, vol. 26, no. 1, pp. 98–119, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. E. Hausenblas and J. Seidler, “A note on maximal inequality for stochastic convolutions,” Czechoslovak Mathematical Journal, vol. 51(126), no. 4, pp. 785–790, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH