Invariant manifolds for weak solutions to stochastic equations
Given stochastic equations \[ d X_t = (A X_t + F(t,X_t)) dt + B(t,X_t) dW_t \] on a separable Hilbert space \(H\) with nonrandom initial conditions \(X_0 = x_0\), where \(W\) denotes a standard Wiener process on some separable Hilbert space \(G\), and \(A\) represents the infinitesimal generator of a strongly continuous semigroup in \(H\). Assume that the random mappings \(F=F(t,\omega,x)\) and \(B=B(t,\omega,x)\) satisfy some appropriate measurability conditions, local Lipschitz and integrability conditions together with some Nagumo type consistency conditions. The author proves a regularity result that any weak solution of these equations, which is viable in a finite-dimensional \(C^2\) submanifold, is a strong solution. The main idea behind the proof steps are the fact that locally any finite-dimensional submanifold \(M\) of \(H\) can be projected diffeomorphically onto a finite-dimensional linear subspace of the domain \(D(A^*)\), where \(A^*\) is the adjoint of \(A\), and an application of the well-known Itô formula. An application of the main theorems to the HJM model for the forward rate curve in mathematical finance is sketched at the end. The main results are related to find finite-dimensional realizations for stochastic equations.
stochastic partial differential equations
scientific article; zbMATH DE number 1562913