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Current Trends in Operator Theory and its Applications pp 493–515Cite as

Birkhäuser

Stochastic Controllability of Linear Interest Rate Models

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 149))

Abstract

We consider the controllability problem for the linear HeathJarrow-Morton-Musiela (HJMM) interest rate model that is realized by an infinite-dimensional stochastic differential equation (SDE). Although it is clear that interest rates are not generally controllable, the objective of our paper is nevertheless to establish necessary and sufficient conditions for the stochastic controllability of a special subclass of the aforementioned models. In this process we determine a control that transfers the said model from an arbitrary interest rate to any other interest rate in the state space of forward rate curves. Our method of solving this problem involves a consideration of the deterministic and stochastic controllability operators related to the aforementioned SDE and their resolvents and a regulator problem associated with the minimum energy principle. In this regard, a formula for a minimizing control is given explicitly in terms of an invertible deterministic controllability operator. Also, we briefly comment on connections between the main results of the paper and the related Ho-Lee, Hull-White and Cox-Ingersoll-Ross interest rate models.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, Potchefstroom University for CHE, Potchefstroom X 6001, South Africa

    Mark A. Petersen

  2. Centre for Business Mathematics and Informatics, North-West University (Potchefstroom Campus), Potchefstroom X 6001, South Africa

    Helgard Raubenheimer, Frederik C. van der Walt & Herrie F. van Rooy II

Authors
  1. Mark A. Petersen
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  2. Helgard Raubenheimer
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  3. Frederik C. van der Walt
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  4. Herrie F. van Rooy II
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Joseph A. Ball  & Martin Klaus  & 

  2. Department of Mathematics, University of California, La Jolla, CA, 92093, USA

    J. William Helton

  3. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Leiba Rodman

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© 2004 Springer Basel AG

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Petersen, M.A., Raubenheimer, H., van der Walt, F.C., van Rooy, H.F. (2004). Stochastic Controllability of Linear Interest Rate Models. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds) Current Trends in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 149. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7881-4_21

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  • DOI: https://doi.org/10.1007/978-3-0348-7881-4_21

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9608-5

  • Online ISBN: 978-3-0348-7881-4

  • eBook Packages: Springer Book Archive

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