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

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JournalFinance and Stochastics
DateAccepted/In press - 3 Jul 2018
DateE-pub ahead of print (current) - 28 Sep 2018
Issue number4
Volume22
Number of pages48
Pages (from-to)959-1006
Early online date28/09/18
Original languageEnglish

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.

    Research areas

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

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