Affine realizations with affine state processes for stochastic partial differential equations
Stefan Tappe
TL;DR
The paper characterizes when semilinear SPDEs with an affine realization admit affine and admissible finite-dimensional state processes by developing an invariant-foliation framework and tangential conditions that decompose the SPDE into a low-dimensional SDE for the state. It establishes necessary and sufficient conditions (including drift-structure α=Sσ^2) for the existence of affine realizations, and constructs maximal sets of initial points from which such realizations can be built. The HJMM forward-rate equation is treated in depth, with explicit one- and two-dimensional realizations and maximal initial-curve sets; the results extend to linear SPDEs and yield concrete examples from natural sciences. Overall, the work provides a rigorous, usable criterion set for obtaining tractable finite-dimensional affine models inside infinite-dimensional SPDEs, with direct relevance to interest-rate modeling and other applied domains.
Abstract
The goal of this paper is to clarify when a stochastic partial differential equation with an affine realization admits affine state processes. This includes a characterization of the set of initial points of the realization. Several examples, as the HJMM equation from mathematical finance, illustrate our results.