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Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations

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  • Zdzisław Brzeźniak

    (University of York)

  • Tayfun Kok

    (University of York)

Abstract

The aim of this paper is threefold. Firstly, we study stochastic evolution equations (with the linear part of the drift being a generator of a C 0 $C_{0}$ -semigroup) driven by an infinite-dimensional cylindrical Wiener process. In particular, we prove, under some sufficient conditions on the coefficients, the existence and uniqueness of solutions for these stochastic evolution equations in a class of Banach spaces satisfying the so-called H $H$ -condition. Moreover, we analyse the Markov property of the solutions. Secondly, we apply the abstract results obtained in the first part to prove the existence and uniqueness of solutions to the Heath–Jarrow–Morton–Musiela (HJMM) equations in weighted Lebesgue and Sobolev spaces. Finally, we study the ergodic properties of the solutions to the HJMM equations. In particular, we find a sufficient condition for the existence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations (when the coefficients are time-independent) in the weighted Lebesgue spaces. Our paper is a modest contribution to the theory of financial models in which the short rate can be undefined.

Suggested Citation

  • Zdzisław Brzeźniak & Tayfun Kok, 2018. "Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations," Finance and Stochastics, Springer, vol. 22(4), pages 959-1006, October.
    • Handle: RePEc:spr:finsto:v:22:y:2018:i:4:d:10.1007_s00780-018-0374-6
    DOI: 10.1007/s00780-018-0374-6
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    References listed on IDEAS

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    1. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    2. Jouini,E. & Cvitanic,J. & Musiela,Marek (ed.), 2001. "Handbooks in Mathematical Finance," Cambridge Books, Cambridge University Press, number 9780521792370, September.
    3. Michael Tehranchi, 2005. "A note on invariant measures for HJM models," Finance and Stochastics, Springer, vol. 9(3), pages 389-398, July.
    4. Irene Klein & Thorsten Schmidt & Josef Teichmann, 2013. "When roll-overs do not qualify as num\'eraire: bond markets beyond short rate paradigms," Papers 1310.0032, arXiv.org.
    5. Carlo Marinelli, 2010. "Well-posedness and invariant measures for HJM models with deterministic volatility and Levy noise," Quantitative Finance, Taylor & Francis Journals, vol. 10(1), pages 39-47.
    6. Tiziano Vargiolu, 1999. "Invariant measures for the Musiela equation with deterministic diffusion term," Finance and Stochastics, Springer, vol. 3(4), pages 483-492.
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    More about this item

    Keywords

    Stochastic evolution equations; Heath–Jarrow–Morton–Musiela equations; Markov semigroup; Invariant measures; Martingale-type 2 Banach spaces;
    All these keywords.

    JEL classification:

    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General

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