×

zbMATH — the first resource for mathematics

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. (English) Zbl 1168.60014
Let \(H\) be a separable Hilbert space and \(A:\, D(A)\subset H\rightarrow H\) be a (in general, unbounded) linear operator generating a pseudo-contraction semigroup. The authors of the present paper are interested in the stochastic differential equation (SDE) \(dZ_t=(AZ_t+B(t,Z))dt+\int_{H\setminus\{0\}}F(t,e,Z)\widetilde{N} (dsde),\, t\in(0,T],\;T>0,\) where \(Z\) is an \(H\)-valued process, and \(\widetilde{N} (dsde)=N(dsde)-dsd\beta(du)\) is a compensated Poisson random measure and \(\beta(de)\) is a Lévy measure on the Borel-\(\sigma\)-field over \(H\setminus\{0\}\). They prove the existence and the uniqueness of a mild solution in the case of coefficients \(B(t,.z),F(t,e,z)\) which can depend on the past of the càdlàd function \(z\) and are Lipschitz in \(z\).
While this existence and uniqueness result was obtained for compensated Poisson random measures associated with canonical Lévy processes on the Skorohod space \(D(R_+,H)\), the authors analyse the case \(B(s,Z)=b(s,Z_s),F(s,e,Z)=f(s,e,Z_s)\) for more general compensated Poisson random measures. In this case they also prove that a Yosida approximation theorem holds, they study the Markov property and they investigate the continuous and also the differentiable dependence of the solution on the initial data. The latter results are got by adapting a method of S. Cerrai. The authors’ paper generalizes in particular former results by Hausenblas (2005) who studied the same type of SDE, but assumed that the operator \(A\) generates a compact analytic semigroup.

MSC:
60H05 Stochastic integrals
60G57 Random measures
46B09 Probabilistic methods in Banach space theory
47G99 Integral, integro-differential, and pseudodifferential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albeverio, S.; Rüdiger, B., The Lévy – ito decomposition theorem and stochastic integrals, on separable Banach spaces, Stoch. anal. appl., 23, 2, 217-253, (2005) · Zbl 1071.60032
[2] Albeverio, S.; Rüdiger, B., Infinite-dimensional stochastic differential equations obtained by subordination and related Dirichlet forms, J. funct. anal., 204, 1, 122-156, (2003) · Zbl 1033.60073
[3] Albeverio, S.; Wu, J.L.; Zhang, T.S., Parabolic SPDEs driven by Poisson white noise, Stochastic process. appl., 74, 1, 21-36, (1998) · Zbl 0934.60055
[4] Applebaum, D., ()
[5] A. Araujo, E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, in: Wiley Series in Probability and Mathematical Statistics, New York, Chichester, Brisbane, Toronto, 1980
[6] Bauer, H., Wahrscheinlichkeitstheorie und grundzüge der masstheorie, 2 edition de gruyter lehrbuch, (1974), Walter de Gruyter Berlin, New York
[7] Bensoussan, A.; Lions, J.-L., Controle impulsionnel et inéquations quasi variationnelles [impulse control and quasivariational inequalities], (), (in French) · Zbl 0491.93002
[8] Bertini, L.; Brassesco, S.; Buttá, P.; Presutti, E., Front fluctuations in one dimensional stochastic phase field equations, Ann. Henri Poincaré, 3, 1, 29-86, (2002) · Zbl 0996.82048
[9] Brzeźniak, Z., Stochastic partial differential equations in \(M\)-type 2 Banach spaces, Potential anal., 4, 1, 1-45, (1995) · Zbl 0831.35161
[10] Brzeźniak, Z., On stochastic convolution in Banach spaces and applications, Stoch. stoch. rep., 61, 3-4, 245-295, (1997) · Zbl 0891.60056
[11] Brzeźniak, Z.; Elworthy, K.D., Stochastic differential equations on Banach manifolds, Methods funct. anal. topology, 6, 1, 43-84, (2000) · Zbl 0965.58028
[12] Cerrai, S., Second order PDE’s in finite and infinite dimension, () · Zbl 0983.60004
[13] Cranston, M.; Le Jan, Y., Self attracting diffusions: two case studies, Math. ann., 303, 1, 87-93, (1995) · Zbl 0838.60052
[14] Dettweiler, E., Banach space valued processes with independent increments and stochastic integrals, (), 54-83 · Zbl 0514.60010
[15] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052
[16] Da Prato, G.; Zabczyk, J., ()
[17] Da Prato, G.; Zabczyk, J., Second order partial differential equations in Hilbert spaces, (), xvi+379 · Zbl 1012.35001
[18] Dellacherie, C.; Meyer, P.A., (), viii+189
[19] Durrett, R.; Rogers, L.C.G., Asymptotic behaviour of Brownian polymers, Probab. theory related. fields, 92, 337-349, (1991) · Zbl 0767.60080
[20] Dynkin, E.B., Die grundlagen der theorie der markoffschen prozesse, (1982), Springer Verlag Berlin, Göttingen, Heidelberg, New York · Zbl 0091.13604
[21] Ethier, S.N.; Kurtz, T.G., Markov processes. characterization and convergence, (), x+534 · Zbl 1089.60005
[22] Filippović, D.; Tappe, S., Existence of Lévy term structure models, Finance stoch., XII, 1, 83-115, (2008) · Zbl 1150.91017
[23] Gawarecki, L.; Mandrekar, V., Stochastic differential equations with discontinuous drift in Hilbert space with applications to interacting particle systems, Proceedings of the seminar on stability problems for stochastic models, part I (nalk eczow, 1999), J. math. sci. (New York), 105, 6, 2550-2554, (2001) · Zbl 0995.60057
[24] Gawarecki, L.; Mandrekar, V.; Richard, P., Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations, Random oper. stoch., 7, 3, 215-240, (1999) · Zbl 0951.60063
[25] Gihman, I.I.; Skorohod, A.V., The theory of stochastic processes II, (1975), Springer Berlin, Heidelberg, New York · Zbl 0305.60027
[26] Hausenblas, E., Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure, Electron. J. probab., 10, 1496-1546, (2005), (electronic) · Zbl 1109.60048
[27] Huang, Z.; Yan, J., Introduction to infinite dimensional stochastic analysis, (), xii+296, Science Press, Beijing, Translated and Revised from the 1997 Chinese edition
[28] Ichikawa, A., Some inequalities for martingales and stochastic convolutions, Stoch. anal. appl., 4, 3, 329-339, (1986) · Zbl 0622.60066
[29] Ikeda, N.; Watanabe, S., North-holland mathematical library, vol. 24, (1989), North Holland Publishing Company Amsterdam, Oxford, New York
[30] Kallianpur, G.; Xiong, J., ()
[31] Knoche, C., SPDEs in infinite dimensional spaces with Poisson noise, C. R. math. acad. sci. Paris, 339, 9, 647-652, (2004) · Zbl 1058.60050
[32] C. Knoche, Existence, Uniqueness and regularity w.r.t. the initial condition of mild solutions to SPDE’s driven by Poisson noise, E05-10-193, preprint BiBoS 2005
[33] C. Knoche, Mild solutions of SPDE’s driven by Poisson noise in infinite dimensions and their dependence on initial data. Doctor-degree Thesis, Fakultät für Mathematik, Universität Bielefeld 2006
[34] Kotelenez, P., A maximal inequality for stochastic convolution integrals on Hilbert space and space time regularity of linear stochastic partial differential equations, Stochastics, 21, 345-458, (1987) · Zbl 0622.60065
[35] Linde, W., Infinitely divisible and stable measures on Banach spaces, () · Zbl 0526.28011
[36] V. Mandrekar, T. Meyer-Brandis, F. Proske, A Bayes formula for non-linear filtering with Gaussian and Cox noise; 2007 (submitted for publication) · Zbl 1234.60041
[37] Mandrekar, V.; Rüdiger, B., Existence and uniqueness of path wise solutions for stochastic integral equations driven by non Gaussian noise on separable Banach spaces, Stochastics, 78, 4, 189-212, (2006) · Zbl 1119.60040
[38] Mandrekar, V.; Rüdiger, B., Lévy noises and stochastic integrals on Banach spaces, (), 193-213 · Zbl 1096.60024
[39] C. Marinelli, Local well-posedness of Musiela’s SPDE with Lévy noise. 2007. arxiv:0704.2380v1 [math.PR.]
[40] Metivier, M., Semimartingales—a course on stochastic processes, De gruyter studies in mathematics, vol. 2, (1982), Berlin New York · Zbl 0503.60054
[41] Metivier, M.; Pellaumail, J., Stochastic integration, () · Zbl 0463.60004
[42] S. Peszat, J. Zabczyk, Heath-Jarrow-Morton-Musiela equation of bond market. Preprint IMPAN 669 Warsaw, 2007
[43] Peszat, S.; Zabczyk, J., Stochastic partial differential equations with Lévy noise, () · Zbl 1205.60122
[44] Protter, P., ()
[45] Rüdiger, B., Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces, Stoch. stoch. rep., 76, 3, 213-242, (2004) · Zbl 1052.60045
[46] Sato, K.I., ()
[47] Skorohod, A.V., Studies in the theory of random processes, (1965), Addison-Wesley Publishing Company, Inc Reading, MA, Translated from the Russian by Scripta Technica, Inc
[48] Yosida, K., Functional analysis, classics in mathematics, reprint of the 1980 edition, (1995), Springer Verlag Berlin, Heidelberg
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.