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Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures. (English) Zbl 1263.60060
Summary: We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called method of the moving frame allows us to reduce the SPDE problems to SDE problems.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[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. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[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. · Zbl 1123.60001
[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. · Zbl 1189.60124 · doi:10.1016/j.jfa.2009.11.006
[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. · Zbl 1228.60002 · doi:10.1007/978-3-642-16194-0
[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. · Zbl 1186.60060 · doi:10.1016/j.jfa.2009.04.015
[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. · Zbl 1230.60066 · doi:10.1080/17442501003624407
[7] P. Kotelenez, “A submartingale type inequality with applications to stochastic evolution equations,” Stochastics, vol. 8, no. 2, pp. 139-151, 1982/83. · Zbl 0495.60066 · doi:10.1080/17442508208833233
[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. · Zbl 0552.60058 · doi:10.1080/07362998408809036
[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. · Zbl 1058.60050 · doi:10.1016/j.crma.2004.09.004
[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. · Zbl 1205.60122 · doi:10.1017/CBO9780511721373
[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. · Zbl 1168.60014 · doi:10.1016/j.spa.2008.03.006
[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. · Zbl 1196.60117 · doi:10.1142/S0219025710003985
[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. · Zbl 1018.60002
[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. · Zbl 1082.60048 · doi:10.1142/S0219493705001584
[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. · Zbl 0321.60056 · numdam:SPS_1975__9__443_0 · eudml:113047
[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. · Zbl 1207.91068 · doi:10.1137/090758593
[19] D. Filipovic, S. Tappe, and J. Teichmann, “Invariant manifolds with boundary for jump diffusions,” http://arxiv.org/abs/1202.1076. · Zbl 1301.60072
[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. · Zbl 0734.60060 · doi:10.1007/978-1-4612-0949-2
[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. · Zbl 1119.60040 · doi:10.1080/17442500600813140
[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. · Zbl 0484.60053
[23] T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,” Journal of Differential Equations, vol. 96, no. 1, pp. 152-169, 1992. · Zbl 0744.34052 · doi:10.1016/0022-0396(92)90148-G
[24] B. Sz.-Nagy and C. Foia\cs, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, The Netherlands, 1970. · Zbl 0201.45003
[25] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London, UK, 1976. · Zbl 0388.46044
[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. · Zbl 1153.60035 · doi:10.1080/07362990701673047
[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. · Zbl 1001.60065 · doi:10.1023/A:1013717013421 · eudml:30671
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