Invariant manifolds with boundary for jump-diffusions
Stefan Tappe (Leibniz Universität Hannover)
Josef Teichmann (ETH Zürich)
Abstract
We provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds with boundary in Hilbert spaces for stochastic partial differential equations driven by Wiener processes and Poisson random measures.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-28
Publication Date: June 22, 2014
DOI: 10.1214/EJP.v19-2882
References
- Abraham, R.; Marsden, J. E.; Ratiu, T. Manifolds, tensor analysis, and applications. Second edition. Applied Mathematical Sciences, 75. Springer-Verlag, New York, 1988. x+654 pp. ISBN: 0-387-96790-7 MR0960687
- Björk, Tomas; Christensen, Bent Jesper. Interest rate dynamics and consistent forward rate curves. Math. Finance 9 (1999), no. 4, 323--348. MR1849252
- Björk, Tomas; Landen, Camilla. On the construction of finite dimensional realizations for nonlinear forward rate models. Finance Stoch. 6 (2002), no. 3, 303--331. MR1914314
- Björk, Tomas; Svensson, Lars. On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance 11 (2001), no. 2, 205--243. MR1822777
- Buckdahn, Rainer; Quincampoix, Marc; Rainer, Catherine; Teichmann, Josef. Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems. Bull. Sci. Math. 134 (2010), no. 2, 207--214. MR2592970
- Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
- Dellacherie, Claude; Meyer, Paul-André. Probabilités et potentiel. (French) Chapitres I à IV. Édition entièrement refondue. Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. XV. Actualités Scientifiques et Industrielles, No. 1372. Hermann, Paris, 1975. x+291 pp. MR0488194
- Filipović, Damir. A note on the Nelson-Siegel family. Math. Finance 9 (1999), no. 4, 349--359. MR1849253
- Filipović, Damir. Exponential-polynomial families and the term structure of interest rates. Bernoulli 6 (2000), no. 6, 1081--1107. MR1809736
- Filipović, Damir. Invariant manifolds for weak solutions to stochastic equations. Probab. Theory Related Fields 118 (2000), no. 3, 323--341. MR1800535
- Filipović, Damir. Consistency problems for Heath-Jarrow-Morton interest rate models. Lecture Notes in Mathematics, 1760. Springer-Verlag, Berlin, 2001. viii+134 pp. ISBN: 3-540-41493-2 MR1828523
- Filipović, Damir; Tappe, Stefan; Teichmann, Josef. Jump-diffusions in Hilbert spaces: existence, stability and numerics. Stochastics 82 (2010), no. 5, 475--520. MR2739608
- Filipović, Damir; Tappe, Stefan; Teichmann, Josef. Term structure models driven by Wiener processes and Poisson measures: existence and positivity. SIAM J. Financial Math. 1 (2010), no. 1, 523--554. MR2669403
- Filipovi'c, D., Tappe, S. and Teichmann, J.: Stochastic partial differential equations and submanifolds in Hilbert spaces. Appendix of phInvariant manifolds with boundary for jump-diffusions, (2014). ARXIV1202.1076v2
- Filipović, Damir; Teichmann, Josef. Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197 (2003), no. 2, 398--432. MR1960419
- Filipović, Damir; Teichmann, Josef. On the geometry of the term structure of interest rates. Stochastic analysis with applications to mathematical finance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 129--167. MR2052259
- Getoor, R. K. On the construction of kernels. Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975), pp. 443--463. Lecture Notes in Math., Vol. 465, Springer, Berlin, 1975. MR0436342
- Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3 MR1943877
- Nakayama, Toshiyuki. Support theorem for mild solutions of SDE's in Hilbert spaces. J. Math. Sci. Univ. Tokyo 11 (2004), no. 3, 245--311. MR2097527
- Nakayama, Toshiyuki. Viability theorem for SPDE's including HJM framework. J. Math. Sci. Univ. Tokyo 11 (2004), no. 3, 313--324. MR2097528
- Rudin, Walter. Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8 MR1157815
- Simon, Thomas. Support theorem for jump processes. Stochastic Process. Appl. 89 (2000), no. 1, 1--30. MR1775224
- Tessitore, Gianmario; Zabczyk, Jerzy. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 (2006), no. 4, 621--655. MR2267702
This work is licensed under a Creative Commons Attribution 3.0 License.