Finite dimensional solutions to SPDEs and the geometry of infinite jet bundles

Abstract

Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A geometrical reformulation of the stochastic problem using the concept of infinite jet bundles is provided and a practical algorithm for explicitly computing these finite dimensional solutions is developed. This method, covering the majority of the current literature, is applied to a set of new SPDEs admitting finite dimensional solutions taken from Heath-Jarrow-Morton framework, stochastic hydrodynamics and filtering theory.

Symmetries of stochastic differential equations using Girsanov transformations

“…The comparison between the determining equations for the Lie point infinitesimal symmetries of the PDE (10) and the determining equation for the stochastic symmetries for the SDE (9) arising in theorem 3.13 is not straightforward, first of all due to the different nature of the involved objects. Indeed, when we look for the infinitesimal symmetries of the PDE we look for a vector field of the form (12), while when we search symmetries for SDE (9), we deal with an infinitesimal transformation V = (Y, C, τ , H), where Y is a vector field on R n+1 , C : R n+1 → so(m), τ : R n+1 → R and H : R n+1 → R m are smooth functions.…”

“…Symmetries of SDEs arising from variational problems has been considered and Noether theorem and integration by quadratures have been generalized to variational stochastic problems (see [34,41,46]). Finally, a generalization of differential constraint method to SPDEs has been proposed in [12,13], while in [2,17] symmetries have been applied to the study of invariant numerical methods for SDEs.…”