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Infinite dimensional affine processes

Thorsten Schmidt, Stefan Tappe, Weijun Yu

TL;DR

This work extends affine diffusion theory to infinite dimensions by formulating affine processes on a Hilbert-space state space, deriving the associated Riccati system, and proving an existence result for strong solutions of infinite-dimensional SDEs with affine coefficients. The authors develop a self-contained infinite-dimensional $Yamada-Watanabe$ framework, employing a retracted subspace with compact embedding to obtain weak solutions and pathwise uniqueness, thereby ensuring a robust affine structure via the Riccati equations. They provide concrete infinite-dimensional examples—Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and Heston-type models—demonstrating how to construct $S$, $m_0$, $M$, and the retracted subspace to satisfy admissibility and invariance conditions. The results enable infinite-factor term-structure modeling and functional-data applications where the factor count is not fixed a priori, with explicit solution forms in several special cases and a clear path to further extensions with Lévy noise.

Abstract

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada-Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

Infinite dimensional affine processes

TL;DR

This work extends affine diffusion theory to infinite dimensions by formulating affine processes on a Hilbert-space state space, deriving the associated Riccati system, and proving an existence result for strong solutions of infinite-dimensional SDEs with affine coefficients. The authors develop a self-contained infinite-dimensional framework, employing a retracted subspace with compact embedding to obtain weak solutions and pathwise uniqueness, thereby ensuring a robust affine structure via the Riccati equations. They provide concrete infinite-dimensional examples—Ornstein-Uhlenbeck, Cox-Ingersoll-Ross, and Heston-type models—demonstrating how to construct , , , and the retracted subspace to satisfy admissibility and invariance conditions. The results enable infinite-factor term-structure modeling and functional-data applications where the factor count is not fixed a priori, with explicit solution forms in several special cases and a clear path to further extensions with Lévy noise.

Abstract

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada-Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

Paper Structure

This paper contains 14 sections, 34 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Infinite dimensional affine processes
  3. Affine processes
  4. Existence of affine processes
  5. Formulation of the existence result
  6. Proof of the existence result
  7. Examples
  8. Particular situations
  9. Infinite dimensional processes of Ornstein-Uhlenbeck type
  10. Infinite dimensional processes of Cox-Ingersoll-Ross type
  11. Infinite dimensional processes of Heston type
  12. Proof of Lemma \ref{['lemma 2.8']}
  13. Infinite dimensional stochastic differential equations
  14. Linear operators in Hilbert spaces

Key Result

Lemma 2.1

For each $x \in {\mathcal{X}}$ the operator $S(x)$ is non-negative definite and trace-class. Moreover, the mapping $\mathop{\mathrm{Tr}}\nolimits S(\cdot):H \to \mathbb{R}$ is continuous.

Theorems & Definitions (74)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Lemma 2.5
  • ...and 64 more