Invariant manifolds for weak solutions to stochastic equations

Abstract

Abstract. Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C 2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C 2 submanifold, is a strong solution.These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.

Invariance of closed convex cones for stochastic partial differential equations

“…(1) Condition (5.8) just means that (a, α) is inward pointing in the sense of Definition C.2. (2) By our stability result for SPDEs (Proposition B.3) and Proposition C. 8 we may assume that α ∈ Lip(H) ∩ F(H).…”

On the Geometry of Interest Rate Models

“…A much more difficult problem is to determine whether any interest rate model is. This is Problem II in Section 3.1 for the NS family, and in a very general setting, inverse consistency problems like this has been studied in great detail by Filipovic in [19], [20], and [21]. In this section we will give an introduction to the Filipovic state space approach to the (inverse) consistency problem, and we will also study a small laboratory example.…”

“…This is of course restrictive. The invariance problem for weak solutions has been studied by Filipovic in [22] and [21]. An alternative way of studying invariance is by using some version of the Stroock-Varadhan support theorem, and this line of thought is carried out in depth in [38].…”

On Invariance of Linear Subspaces for Stochastic Evolution Equations

“…The set of positive functions in L 2 is characterized by Tessitore and Zabczyk (1998). Polyhedrons in Hilbert spaces are characterized by Milian (1998).The invariance problem with respect to weak solutions to (1), related to finitedimensional submanifolds in H is studied by Filipovic (1999). Compared to result by Jachimiak, we consider linear subspaces of H however our assumptions on coefficients are more general.…”