Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equations: HJMM equations in weighted $L^p$ spaces

Zdzislaw Brzezniak; Tayfun Kok

doi:10.1007/s00780-018-0374-6

Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equations: HJMM equations in weighted $L^p$ spaces

Zdzislaw Brzezniak, Tayfun Kok

Mathematics

Research output: Contribution to journal › Article › peer-review

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 $C_0$-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 $H$-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 the 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.

Original language	English
Pages (from-to)	959-1006
Number of pages	48
Journal	Finance and Stochastics
Volume	22
Issue number	4
Early online date	28 Sept 2018
DOIs	https://doi.org/10.1007/s00780-018-0374-6
Publication status	E-pub ahead of print - 28 Sept 2018

Keywords

stochastic evolution equations
Heath-Jarrow-Morton-Musiela equations
Markov semigroup
invariant measures
martingale type 2 Banach spaces

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10.1007/s00780-018-0374-6Licence: Other

Brzeźniak-Kok2018_Article_StochasticEvolutionEquationsIn
© The Author(s) 2018.
Final published version, 1.04 MBLicence: CC BY

Cite this

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title = "Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equations: HJMM equations in weighted $L^p$ spaces",

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 $C_0$-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 $H$-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 the 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.",

keywords = "stochastic evolution equations, Heath-Jarrow-Morton-Musiela equations , Markov semigroup , invariant measures , martingale type 2 Banach spaces",

author = "Zdzislaw Brzezniak and Tayfun Kok",

year = "2018",

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doi = "10.1007/s00780-018-0374-6",

language = "English",

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T1 - Stochastic evolution equations in Banach spaces and applications to the Heath-Jarrow-Morton-Musiela equations

T2 - HJMM equations in weighted $L^p$ spaces

AU - Brzezniak, Zdzislaw

AU - Kok, Tayfun

PY - 2018/9/28

Y1 - 2018/9/28

N2 - 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 $C_0$-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 $H$-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 the 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.

AB - 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 $C_0$-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 $H$-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 the 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.

KW - stochastic evolution equations

KW - Heath-Jarrow-Morton-Musiela equations

KW - Markov semigroup

KW - invariant measures

KW - martingale type 2 Banach spaces

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DO - 10.1007/s00780-018-0374-6

M3 - Article

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JF - Finance and Stochastics

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