Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties
Abstract
We use the Yosida approximation to find an Itô formula for mild solutions
Acknowledgements
We are very grateful to Prof. P. Sundar (Louisiana State University) for useful discussions related to this work.
References
[1] S. Albeverio, V. Mandrekar and B. Rüdiger, Existence of mild solutions for SDE’s and semilinear equations with non-Gaussian noise, Stochastic Process. Appl. 119 (2009), 835–863. 10.1016/j.spa.2008.03.006Suche in Google Scholar
[2] S. Albeverio and B. Rüdiger, Stochastic integrals and the Lévy-Ito decomposition theorem on separable Banach spaces, Stoch. Anal. Appl. 23 (2005), no. 2, 217–253. 10.1081/SAP-200026429Suche in Google Scholar
[3] D. Applebaum, Lévy Process and Stochastic Calculus, 2nd ed., Cambridge University Press, Cambridge, 2009. 10.1017/CBO9780511809781Suche in Google Scholar
[4] Z. Brzeźniak, E. Hausenblas and J. Zhu, Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces, A mild Itô formula for SPDEs, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 2, 937–956. 10.1214/16-AIHP743Suche in Google Scholar
[5] G. Da Prato, A. Jentzen and M. Röckner, preprint (2011), https://arxiv.org/abs/1009.3526v3. Suche in Google Scholar
[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. 10.1017/CBO9780511666223Suche in Google Scholar
[7] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, John Wiley & Sons, New York, 1986. 10.1002/9780470316658Suche in Google Scholar
[8] G. Fabbri and F. Russo, Infinite dimensional weak Dirichlet processes and convolution type processes, Stochastic Process. Appl. 127 (2017) no. 1, 325–357. 10.1016/j.spa.2016.06.010Suche in Google Scholar
[9] D. Fillipović, S. Tappe and J. Teichmann, Term structure models driven by Wiener process and Poisson measures: Existence and positivity, SIAM J. Financial Math. 1 (2010), no. 1, 523–554. 10.1137/090758593Suche in Google Scholar
[10] L. Gawarecki and V. Mandrekar, Stochastic Differential Equation in Infinite Dimension, Springer, Heidelberg, 2011. 10.1007/978-3-642-16194-0Suche in Google Scholar
[11] A. Ichikawa, Stability of semilinear stochastic evolution equations, Math. Anal. Appl. 90 (1982), 12–44. 10.1016/0022-247X(82)90041-5Suche in Google Scholar
[12] A. Ichikawa, Semilinear stochastic evolution equations: Boundedness, stability and invariant measures, Stochastics 12 (1984), 1–39. 10.1080/17442508408833293Suche in Google Scholar
[13] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1989. Suche in Google Scholar
[14] V. Mandrekar and B. Rüdiger, Stochastic Integration in Banach Spaces, Springer, Cham, 2015. 10.1007/978-3-319-12853-5Suche in Google Scholar
[15] V. Mandrekar, B. Rüdiger and S. Tappe, Ito’s formula for Banach space valued jump processes driven by Poisson random measures, Seminar on Stochastic Analysis, Random Fields and Applications VII (Ascona 2011), Progr. Probab. 67, Birkhäuser, Basel (2013), 171–186. 10.1007/978-3-0348-0545-2_7Suche in Google Scholar
[16] V. Mandrekar and L. Wang, Asymptotic properties of SPDE in Hilbert spaces driven by non-Gaussian noise, Commun. Stoch. Anal. 5 (2011), no. 2, 309–331. 10.31390/cosa.5.2.04Suche in Google Scholar
[17] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Cambridge University Press, Cambridge, 2007. 10.1017/CBO9780511721373Suche in Google Scholar
[18] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, Berlin, 1999. 10.1007/978-3-662-06400-9Suche in Google Scholar
[19] B. Rüdiger, Stochastic integration with respect to Poisson random measure on seperable Banach space, Stoch. Stoch. Rep. 76 (2004), 213–242. 10.1080/10451120410001704081Suche in Google Scholar
[20] B. Rüdiger and G. Ziglio, Ito formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces, Stochastics 78 (2006), no. 6, 377–410. 10.1080/17442500600976137Suche in Google Scholar
[21] A. V. Skorokhod, Studies in the Theory of Random Processes, Addison-Wesley, Reading, 1965. Suche in Google Scholar
[22] B. Sz.-Nagy, Sur les contractions de l’espace de Hilbert, Acta Sci. Math. (Szeged) 15 (1953), 87–92. Suche in Google Scholar
[23] B. Sz.-Nagy, Transfotmations de l’espace de Hilbert, functions de type positif sur un groupe, Acta Sci. Math. (Szeged) 15 (1954), 104–114. Suche in Google Scholar
[24] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. Suche in Google Scholar
[25] J. Zhu and Z. Brzeźniak, Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 9, 3269–3299. 10.3934/dcdsb.2016097Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- The distribution of random motion at non-constant velocity in semi-Markov media
- Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties
- New exact solutions for the Wick-type stochastic Zakharov–Kuznetsov equation for modelling waves on shallow water surfaces
- On the multi-dimensional Favard lemma
Artikel in diesem Heft
- Frontmatter
- The distribution of random motion at non-constant velocity in semi-Markov media
- Itô formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties
- New exact solutions for the Wick-type stochastic Zakharov–Kuznetsov equation for modelling waves on shallow water surfaces
- On the multi-dimensional Favard lemma