Existence of affine realizations for stochastic partial differential equations driven by Lévy processes

TL;DR

The paper characterizes when a semilinear SPDE driven by Lévy processes admits an affine realization, i.e., a finite-dimensional invariant foliation that renders the dynamics tractable. It develops necessary and sufficient conditions in terms of $A$-invariance, the drift projection, and volatility containment, and extends the analysis to linear SPDEs via $A$-quasi-exponential volatility, including a constructive framework via operator transformations. The results are applied to the HJMM equation and a collection of linear SPDEs, with rigorous proofs for the foliation-based approach and clear criteria for the linear case. This provides a principled method to reduce infinite-dimensional Lévy-driven SPDEs to finite-dimensional dynamics, enabling tractable analysis in mathematical finance and applied sciences.

Abstract

The goal of this paper is to clarify when a semilinear stochastic partial differential equation driven by Lévy processes admits an affine realization. Our results are accompanied by several examples arising in natural sciences and economics.

Theorems & Definitions (57)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1