Stochastic invariance and consistency of financial models
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 2, pp. 67-80.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

The paper is devoted to a connection between stochastic invariance in infinite dimensions and a consistency question of mathematical finance. We derive necessary and sufficient conditions for stochastic invariance of Nagumo’s type for stochastic equations with additive noise. They are applied to Ornstein-Uhlenbeck processes and to specific financial models. The case of evolution equations with general noise is discussed also and a comparison with recent results obtained by geometric methods is presented as well.
Questo lavoro riguarda la connessione fra l’invarianza stocastica in dimensione infinita e un problema di consistenza in finanza matematica. Vengono date condizioni necessarie e sufficienti di tipo Nagumo per l’invarianza di equazioni stocastiche con rumore additivo. Esse sono applicate a processi di Ornstein-Uhlenbeck e specifici modelli finanziari. Vengono anche discusse equazioni di evoluzione con rumore generale e viene fatto un paragone con recenti risultati ottenuti con metodi geometrici.
@article{RLIN_2000_9_11_2_a0,
     author = {Zabczyk, Jerzy},
     title = {Stochastic invariance and consistency of financial models},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {67--80},
     publisher = {mathdoc},
     volume = {Ser. 9, 11},
     number = {2},
     year = {2000},
     zbl = {0978.60039},
     mrnumber = {1134779},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_2_a0/}
}
TY  - JOUR
AU  - Zabczyk, Jerzy
TI  - Stochastic invariance and consistency of financial models
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2000
SP  - 67
EP  - 80
VL  - 11
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_2_a0/
LA  - en
ID  - RLIN_2000_9_11_2_a0
ER  - 
%0 Journal Article
%A Zabczyk, Jerzy
%T Stochastic invariance and consistency of financial models
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2000
%P 67-80
%V 11
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_2_a0/
%G en
%F RLIN_2000_9_11_2_a0
Zabczyk, Jerzy. Stochastic invariance and consistency of financial models. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 11 (2000) no. 2, pp. 67-80. http://geodesic.mathdoc.fr/item/RLIN_2000_9_11_2_a0/

[1] J. P. Aubin, Viability Theory. Birkhäuser, Boston-Basel 1991. | MR | Zbl

[2] J. P. Aubin - G. Da Prato, The viability theorem for stochastic differential inclusions. Stochastic Analysis and Applications, 16, 1998, 1-15. | DOI | MR | Zbl

[3] V. Bally - A. Millet - M. Sanz-Sole, Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations. Annals of Probability, 23, 1995, 178-222. | fulltext mini-dml | MR | Zbl

[4] T. Björk - B. J. Christensen, Interest rate dynamics and consistent forward rate curves. Math. Finance, to appear. | DOI | MR | Zbl

[5] T. Björk - A. Gombani, Minimal realizations of forward rates. Finance and Stochastics, vol. 3, 1999, 423-432.

[6] T. Björk - G. Di Masi - Yu. Kabanov - W. Runggaldier, Towards a general theory of bond markets. Finance and Stochastic, vol. 1, 1996, 141-174. | MR | Zbl

[7] T. Björk - Lars Svensson, On the existence of finite dimensional realizations for nonlinear forward rate models. Preprint, March 1999. | Zbl

[8] A. Brace - D. Gątarek - M. Musiela, The market model of interest rate dynamics. Math. Finance, 7, 1997, 127-154. | DOI | MR | Zbl

[9] A. Brace - M. Musiela, A multifactor Gauss Markov implementation of Heath, Jarrow and Morton. Math. Finance, 4, 1994, 259-283. | Zbl

[10] J. P. Bouchaud - R. Cont - N. El Karoui - M. Potters - N. Sagna, Phenomenology of the interest rate curve: a statistical analysis of the term structure deformations. Working paper, 1997. (http:= =econwpa.wustl.edu/ewp-fin/9712009). | Zbl

[11] R. Cont, Modelling term structure dynamics: an infinite dimensional approach. International Journal of Theoretical and Applied Finance, to appear. | DOI | MR | Zbl

[12] G. Da Prato - J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge-New York 1992. | DOI | MR | Zbl

[13] G. Da Prato - J. Zabczyk, Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge-New York 1996. | DOI | MR | Zbl

[14] D. Filipovic, Invariant Manifolds for Weak Solutions to Stochastic Equations. Manuscript, March 1999. | DOI | MR | Zbl

[15] D. Gątarek, Some remarks on the market model of interest rates. Control and Cybernetics, vol. 25, 1996, 1233-1244. | MR | Zbl

[16] D. Heath - R. Jarrow - A. Morton, Bond pricing and the term structure of interest rates: a new methodology. Econometrica, 60, 1992, 77-101. | Zbl

[17] W. Jachimiak, A note on invariance for semilinear differential equations. Bull. Pol. Sci., 44, 1996, 179-183. | MR | Zbl

[18] W. Jachimiak, Invariance problem for evolution equations. PhD Thesis, Institute of Mathematics Polish Academy of Sciences, Warsaw 1998 (in Polish).

[19] W. Jachimiak, Stochastic invariance in infinite dimensions. Preprint 591, Institute of Mathematics Polish Academy of Sciences, Warsaw, October 1998.

[20] A. Milian, Nagumo’s type theorems for stochastic equations. PhD Thesis, Institute of Mathematics Polish Academy of Sciences, 1994.

[21] A. Milian, Invariance for stochastic equations with regular coefficients. Stochastic Analysis and Applications, 15, 1997, 91-101. | DOI | MR | Zbl

[22] A. Millet - M. Sanz-Sole, The support of the solution to a hyperbolic spde. Probab. Th. Rel. Fields, 84, 1994, 361-387. | DOI | MR | Zbl

[23] A. Millet - M. Sanz-Sole, Approximation and support theorem for a two space-dimensional wave equations. Mathematical Sciences Research Institute, Preprint No. 1998-020, Berkeley, California. | fulltext mini-dml

[24] M. Musiela, Stochastic PDEs and term structure models. Journees International de Finance, IGR-AFFI, La Baule 1993.

[25] M. Musiela - M. Rutkowski, Martingale Methods in Financial Modelling. Applications of Mathematics, vol. 36, Springer-Verlag, 1997. | MR | Zbl

[26] N. Pavel, Invariant sets for a class of semilinear equations of evolution. Nonl. Anal. Theor., 1, 1977, 187-196. | MR | Zbl

[27] N. Pavel, Differential equations, flow invariance. Pitman Lecture Notes, Boston 1984.

[28] G. Tessitiore - J. Zabczyk, Comments on transition semigroups and stochastic invariance. Scuola Normale Superiore, Pisa 1998, preprint. | MR

[29] K. Twardowska, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations. Stoch. Anal. Appl., 13, 1995, 601-626. | DOI | MR | Zbl

[30] K. Twardowska, Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. Dissertationes Mathematicae, CCCXXV, 1993. | MR | Zbl

[31] D. W. Stroock - S. R. S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle. Proceedings 6th Berkeley Symposium Math. Statist. Probab., vol. 3, University of California Press, Berkeley 1972, 333-359. | fulltext mini-dml | MR | Zbl

[32] K. Yosida, Functional Analysis. Springer-Verlag, 1965.

[33] J. Zabczyk, Mathematical Control Theory: An Introduction. Birkhäuser, Boston 1992. | MR | Zbl

[34] J. Zabczyk, Stochastic invariance and consistency of financial models. Preprints di Matematica n. 7, Scuola Normale Superiore, Pisa 1999. | MR | Zbl