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Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

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  • Albeverio, S.
  • Mandrekar, V.
  • Rüdiger, B.

Abstract

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.

Suggested Citation

  • Albeverio, S. & Mandrekar, V. & Rüdiger, B., 2009. "Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 835-863, March.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:3:p:835-863
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    References listed on IDEAS

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    1. Damir Filipović & Stefan Tappe, 2008. "Existence of Lévy term structure models," Finance and Stochastics, Springer, vol. 12(1), pages 83-115, January.
    2. Albeverio, Sergio & Wu, Jiang-Lun & Zhang, Tu-Sheng, 1998. "Parabolic SPDEs driven by Poisson white noise," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 21-36, May.
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    Cited by:

    1. Boufoussi, Brahim & Hajji, Salah, 2010. "Successive approximation of neutral functional stochastic differential equations with jumps," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 324-332, March.
    2. Yang, Xu & Zhao, Weidong, 2018. "Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 58-75.
    3. Y. Ren & Q. Zhou & L. Chen, 2011. "Existence, Uniqueness and Stability of Mild Solutions for Time-Dependent Stochastic Evolution Equations with Poisson Jumps and Infinite Delay," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 315-331, May.
    4. Albeverio, Sergio & Mastrogiacomo, Elisa & Smii, Boubaker, 2013. "Small noise asymptotic expansions for stochastic PDE’s driven by dissipative nonlinearity and Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2084-2109.
    5. Albeverio, Sergio & Smii, Boubaker, 2015. "Asymptotic expansions for SDE’s with small multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 1009-1031.

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