Invariance of closed convex cones for stochastic partial differential equations

TL;DR

This work characterizes when a closed convex cone $K\subset H$ remains invariant for a jump-diffusion SPDE driven by a Wiener process and a Poisson random measure. It derives necessary and sufficient geometric conditions on the coefficients $A$, $\alpha$, $\sigma$, and $\gamma$, expressed via jump-invariance, inward-pointing drift, parallel boundary diffusion, and cone-invariance of the semigroup, under mild regularity and Schauder-basis assumptions. The results are built through a layered approach: necessity via short-time analysis, sufficiency first for diffusion with smooth volatilities, then Lipschitz volatilities, and finally general jump-diffusion with stability-based approximation arguments. An explicit example on $\ell^2$ demonstrates the applicability to order- and positivity-preserving SPDE models. Overall, the paper provides a complete equivalence framework for stochastic cone invariance useful in stability and financial-modeling contexts.

Abstract

The goal of this paper is to clarify when a closed convex cone is invariant for a stochastic partial differential equation (SPDE) driven by a Wiener process and a Poisson random measure, and to provide conditions on the parameters of the SPDE, which are necessary and sufficient.

Figures (2)

• Figure 1: Illustration of the invariance conditions.
• Figure 2: Approximation with locally parallel functions.

Theorems & Definitions (143)

• Theorem 1.1
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Lemma 2.10
• proof