Flatness of invariant manifolds for stochastic partial differential equations driven by Lévy processes

Stefan Tappe

doi:10.1214/ecp.v20-3943

Abstract

The purpose of this note is to prove that the flatness of an invariant manifold for a semilinear stochastic partial differential equation driven by Levy processes is at least equal to the number of driving sources with small jumps. We illustrate our findings by means of an example.

Highlights

The purpose of this note is to prove that the flatness of an invariant manifold for a semilinear stochastic partial differential equation driven by Lévy processes is at least equal to the number of driving sources with small jumps
The purpose of this note is to show that an invariant manifold for a semilinear stochastic partial differential equation (SPDE)
A result which is related to the findings of our paper has been provided in [9] for the particular case of Wiener process driven Heath-Jarrow-Morton (HJM, see [10]) interest rate term structure models, namely that under suitable conditions an invariant manifold for the HJM equation necessarily is a foliation, that is, a collection of affine spaces

Summary

Introduction

The purpose of this note is to show that an invariant manifold for a semilinear stochastic partial differential equation (SPDE). We deal with general SPDEs of the type (1.1) driven by Lévy processes, and the intuitive statement of our main results (Theorems 2.6 and 2.7) is that the flatness of an invariant manifold is at least equal to the number of driving sources with small jumps. Since the manifold M is invariant, it captures every possible jump of Xk. Since, in addition, the Lévy process Xk makes arbitrary small positive jumps, this means that for some > 0 we have h + xkγk(h) ∈ M for all xk ∈ [0, ]. Due to the previous step, the linear space L generated by all γk(h), k ∈ K is contained in the tangent space to M at h, which provides the desired result concerning the flatness of the manifold. For convenience of the reader, in Appendix A we provide the crucial definitions and results regarding submanifolds in Hilbert spaces

Flatness of invariant manifolds

An example

A Finite dimensional submanifolds in Hilbert spaces