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Stochastic Evolution Equations in Banach Spaces and Applications to Heath-Jarrow-Morton-Musiela Equation

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  • Zdzislaw Brzezniak
  • Tayfun Kok

Abstract

In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM) equation (6.3). In particular, we prove the existence and the uniqueness of solutions to the latter equation in the weighted Lebesgue and Sobolev spaces respectively. We also find a sufficient condition for the existence and the uniqueness of an invariant measure for the Markov semigroup associated to equation (6.3) in the weighted Lebesgue spaces.

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  • Zdzislaw Brzezniak & Tayfun Kok, 2016. "Stochastic Evolution Equations in Banach Spaces and Applications to Heath-Jarrow-Morton-Musiela Equation," Papers 1608.05814, arXiv.org.
    • Handle: RePEc:arx:papers:1608.05814
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    References listed on IDEAS

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    1. Tiziano Vargiolu, 1999. "Invariant measures for the Musiela equation with deterministic diffusion term," Finance and Stochastics, Springer, vol. 3(4), pages 483-492.
    2. 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..
    3. Damir Filipović & Stefan Tappe, 2008. "Existence of Lévy term structure models," Finance and Stochastics, Springer, vol. 12(1), pages 83-115, January.
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