The Annals of Applied Probability

Hypoellipticity in infinite dimensions and an application in interest rate theory

Fabrice Baudoin and Josef Teichmann

Full-text: Open access


We apply methods from Malliavin calculus to prove an infinite-dimensional version of Hörmander’s theorem for stochastic evolution equations in the spirit of Da Prato–Zabczyk. This result is used to show that HJM-equations from interest rate theory, which satisfy the Hörmander condition, have the conceptually undesirable feature that any selection of yields admits a density as multi-dimensional random variable.

Article information

Ann. Appl. Probab. Volume 15, Number 3 (2005), 1765-1777.

First available in Project Euclid: 15 July 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Generic evolutions in interest rate theory HJM equations Hörmander’s theorem Malliavin calculus hypoellipticity


Baudoin, Fabrice; Teichmann, Josef. Hypoellipticity in infinite dimensions and an application in interest rate theory. Ann. Appl. Probab. 15 (2005), no. 3, 1765--1777. doi:10.1214/105051605000000214.

Export citation


  • Airault, H. (1989). Projection of the infinitesimal generator of a diffusion. J. Funct. Anal. 85 353--391.
  • Bismut, J. M. (1981). Martingales, the Malliavin calculus, and hypoellipticity under general Hörmander conditions. Z. Wahrsch. Verw. Gebiete 56 469--505.
  • Björk, T. and Svensson, L. (2001). On the existence of finite-dimensional realizations for nonlinear forward rate models. Math. Finance 11 205--243.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • Filipović, D. (2001). Consistency Problems for Heath--Jarrow--Morton Interest Rate Models. Springer, Berlin.
  • Filipović, D. and Teichmann, J. (2003). Existence of invariant manifolds for stochastic equations in infinite dimension. J. Funct. Anal. 197 398--432.
  • Hairer, M., Mattingly, J. and Pardoux, É. (2004). Malliavin calculus for highly degenerate 2D stochastic Navier--Stokes equations. C. R. Math. Acad. Sci. Paris 339 793--796.
  • Hubalek, F., Klein, I. and Teichmann, J. (2002). A general proof of the Dybvig--Ingersoll--Ross theorem: Long forward rates can never fall. Math. Finance 12 447--451.
  • Malliavin, P. (1978). Stochastic calculus of variations and hypoelliptic operators. In Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976 195--263. Wiley, New York.
  • Malliavin, P. (1997). Stochastic Analysis. Springer, Berlin.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, Berlin.
  • Ocone, D. (1988). Stochastic calculus of variations for stochastic partial differential equations. J. Funct. Anal. 79 288--331.