Existence of invariant manifolds for stochastic equations in infinite dimension
The authors investigate the existence of finite-dimensional invariant manifolds for a stochastic equation of the type \[ dr_t= \bigl(Ar_t+ \alpha(r_t) \bigr)dt+ \sum^d_{j=1} j(r_t)dW^j_t, \quad r_0=h_0, \] on a separable Hilbert space \(H\), in the spirit of \textit{G. Da Prato} and \textit{J. Zabczyk} [``Stochastic equations in infinite dimensions'' (1992; Zbl 0761.60052)]. The operator \(A:D(A) \subset H\to H\) generates a strongly continuous semigroup on \(H\); here \(d\in\mathbb{N}\), and \(W=(W^1,\dots,W^d)\) denotes a standard \(d\)-dimensional Brownian motion, the mappings \(\alpha:H\to H\) and \(\sigma= (\sigma_1, \dots,\sigma_d):\) \(H\to H^d\) satisfy a smoothness condition. The main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application, the authors characterize all finite-dimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates.
scientific article; zbMATH DE number 1895828