×

zbMATH — the first resource for mathematics

Existence of invariant manifolds for stochastic equations in infinite dimension. (English) Zbl 1013.60035
The authors investigate the existence of finite-dimensional invariant manifolds for a stochastic equation of the type \[ dr_t= \bigl(Ar_t+ \alpha(r_t) \bigr)dt+ \sum^d_{j=1} j(r_t)dW^j_t, \quad r_0=h_0, \] on a separable Hilbert space \(H\), in the spirit of G. Da Prato and J. Zabczyk [“Stochastic equations in infinite dimensions” (1992; Zbl 0761.60052)]. The operator \(A:D(A) \subset H\to H\) generates a strongly continuous semigroup on \(H\); here \(d\in\mathbb{N}\), and \(W=(W^1,\dots,W^d)\) denotes a standard \(d\)-dimensional Brownian motion, the mappings \(\alpha:H\to H\) and \(\sigma= (\sigma_1, \dots,\sigma_d):\) \(H\to H^d\) satisfy a smoothness condition. The main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application, the authors characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34G25 Evolution inclusions
37L55 Infinite-dimensional random dynamical systems; stochastic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Björk, T.; Christensen, B. J.: Interest rate dynamics and consistent forward rate curves. Math. finance 9, 323-348 (1999) · Zbl 0980.91030
[2] Björk, T.; Landén, C.: On the construction of finite dimensional realizations for nonlinear forward rate models. Finance stochast. 6, 303-331 (2002) · Zbl 1026.60084
[3] Björk, T.; Svensson, L.: On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. finance 11, 205-243 (2001) · Zbl 1055.91017
[4] Cox, J.; Ingersoll, J.; Ross, S.: A theory of the term structure of interest rates. Econometrica 53, 385-408 (1985) · Zbl 1274.91447
[5] Da Prato, G.; Zabczyk, J.: Stochastic equations in infinite dimensions. (1992) · Zbl 0761.60052
[6] D. Duffie, D. Filipović, W. Schachermayer, Affine processes and applications in finance, Working paper, 2001. · Zbl 1048.60059
[7] Duffie, D.; Kan, R.: A yield-factor model of interest rates. Math. finance 6, 379-406 (1996) · Zbl 0915.90014
[8] Filipović, D.: Consistency problems for heath–jarrow–morton interest rate models. (2001) · Zbl 1008.91038
[9] Filipović, D.: Exponential-polynomial families and the term structure of interest rates. Bernoulli 6, 1-27 (2000) · Zbl 0982.60085
[10] Filipović, D.: Invariant manifolds for weak solutions to stochastic equations. Probab. theory relat. Fields 118, 323-341 (2000) · Zbl 0970.60069
[11] D. Filipović, J. Teichmann, Regularity of finite-dimensional realizations for evolution equations, J. Funct. Anal. · Zbl 1011.37004
[12] D. Filipović, J. Teichmann, On the geometry of the term structure of interest rates, submitted for publication. · Zbl 1048.60045
[13] Hamilton, R. S.: The inverse function theorem of Nash and Moser. Bull. amer. Math. soc. 7, 65-222 (1982) · Zbl 0499.58003
[14] Heath, D.; Jarrow, R.; Morton, A.: Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60, 77-105 (1992) · Zbl 0751.90009
[15] Kolář, I.; Michor, P. W.; Slovák, J.: Natural operations in differential geometry. (1993) · Zbl 0782.53013
[16] A. Kriegl, P.W. Michor, The Convenient Setting for Global Analysis, Surveys and Monographs, Vol. 53, Amer. Math. Soc., Providence, RI, 1997. · Zbl 0889.58001
[17] Lang, S.: Fundamentals of differential geometry, graduate texts in mathematics. 191 (1999)
[18] S. G. Lobanov, O.G. Smolyanov, Ordinary differential equations in locally convex spaces, Russian Math. Surv. London Math. Soc., (1993) 97–175. · Zbl 0834.34076
[19] M. Musiela, Stochastic PDEs and term structure models, Journées Internationales de Finance, IGR-AFFI, La Baule, 1993.
[20] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194, Springer, Berlin, New York, Tokyo, 1986. · Zbl 0616.34060
[21] L.E.O. Svensson, Estimating and interpreting forward interest rates: Sweden 1992–1994, IMF Working Paper No. 114, September 1994.
[22] J. Teichmann, Hille–Yosida theory in convenient analysis, Rev. Math. Complutense forthcoming (2002). · Zbl 1041.47025
[23] Teichmann, J.: A Frobenius theorem on convenient manifolds. Monatsh. math. 134, 159-167 (2001) · Zbl 1004.58002
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.