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

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 C0-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 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.