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

Zdzisław Brzeźniak () and Tayfun Kok ()
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Zdzisław Brzeźniak: University of York
Tayfun Kok: University of York

Finance and Stochastics, 2018, vol. 22, issue 4, No 6, 959-1006

Abstract: 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.

Keywords: Stochastic evolution equations; Heath–Jarrow–Morton–Musiela equations; Markov semigroup; Invariant measures; Martingale-type 2 Banach spaces; 60H15; 91G80; 37L40; 37L55; 35R10; 47D06; 91B24; 91B25; 91B55; 91B70 (search for similar items in EconPapers)
JEL-codes: C60 E43 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s00780-018-0374-6

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