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Invariant cones for jump-diffusions in infinite dimensions

Stefan Tappe

TL;DR

The paper develops robust invariance criteria for closed convex cones in infinite-dimensional SPDEs with jump-diffusion, relaxing the previous requirement that cones be generated by an unconditional Schauder basis to an approximate-generation notion, and providing a simplified drift condition. It proves diffusion and general jump-diffusion invariance theorems, with additional results on isomorphic and product-cone transformations, and extends these results to both standard and abstract $L^2$-spaces. The framework encompasses self-dual and $L^2$-type cones, and the authors illustrate the theory across a broad set of applications in natural sciences and economics, including HJMM forward-rate dynamics, stochastic cable/heat equations, and energy/credit markets. Collectively, these results offer practical, mathematically rigorous tools to enforce positivity and monotonicity constraints in high-dimensional SPDE models with jumps. The work thereby enhances the reliability and scope of positivity-preserving modeling in infinite-dimensional stochastic systems.

Abstract

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. Our results apply to the positive cone of abstract $L^2$-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

Invariant cones for jump-diffusions in infinite dimensions

TL;DR

The paper develops robust invariance criteria for closed convex cones in infinite-dimensional SPDEs with jump-diffusion, relaxing the previous requirement that cones be generated by an unconditional Schauder basis to an approximate-generation notion, and providing a simplified drift condition. It proves diffusion and general jump-diffusion invariance theorems, with additional results on isomorphic and product-cone transformations, and extends these results to both standard and abstract -spaces. The framework encompasses self-dual and -type cones, and the authors illustrate the theory across a broad set of applications in natural sciences and economics, including HJMM forward-rate dynamics, stochastic cable/heat equations, and energy/credit markets. Collectively, these results offer practical, mathematically rigorous tools to enforce positivity and monotonicity constraints in high-dimensional SPDE models with jumps. The work thereby enhances the reliability and scope of positivity-preserving modeling in infinite-dimensional stochastic systems.

Abstract

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. Our results apply to the positive cone of abstract -spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.
Paper Structure (14 sections, 79 theorems, 244 equations)

This paper contains 14 sections, 79 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Closed convex cones in Hilbert spaces
  3. Self-dual cones in Hilbert space
  4. The invariance result for diffusion SPDEs
  5. The invariance result for general jump-diffusion SPDEs
  6. Isomorphic transformations of closed convex cones
  7. Products of closed convex cones
  8. Isometric transformations of products of closed convex cones
  9. $L^2$-spaces
  10. Abstract $L^2$-spaces
  11. Examples of Hilbert spaces and semigroups
  12. Applications
  13. Generalized Filipović spaces
  14. Orthogonal projections in $L^2$-spaces

Key Result

Lemma 2.3

Let $G \subset H$ be a subset and set $F := \overline{{\rm lin}}^+ G$. Then we have $G^* = F^*$.

Theorems & Definitions (179)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 169 more