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Information geometry of tempered stable processes

Jaehyung Choi

TL;DR

The paper addresses the information geometry of tempered stable processes by deriving the $D^{(α)}$ divergence between processes and, from it, the Fisher information metric and $α$-connections on the statistical manifolds of CTS, GTS, and RDTS distributions. It demonstrates that the geometry is notably tractable: for GTS, CTS, and RDTS the Fisher information matrices are diagonal in natural tail-parameter coordinates and the $α$-connections are explicit, yielding $e$-flat manifolds. These results are then leveraged for practical statistical applications, including bias reduction via Jeffreys priors and construction of Komaki-style Bayesian predictive priors, using the common geometric structure across tempered stable processes. Overall, the framework generalizes previous work on CTS relative entropy to a broader class of tempered stable models and provides concrete tools for inference in heavy-tailed time-series contexts. This contributes a rigorous, operationally useful information-geometric toolkit for tempering stable processes in finance and related fields.

Abstract

We find the information geometry of tempered stable processes. Beginning with the derivation of $α$-divergence between two tempered stable processes, we obtain the corresponding Fisher information matrices and the $α$-connections on their statistical manifolds. Furthermore, we explore statistical applications of this geometric framework. Various tempered stable processes such as generalized tempered stable processes, classical tempered stable processes, and rapidly-decreasing tempered stable processes are presented as illustrative examples.

Information geometry of tempered stable processes

TL;DR

The paper addresses the information geometry of tempered stable processes by deriving the divergence between processes and, from it, the Fisher information metric and -connections on the statistical manifolds of CTS, GTS, and RDTS distributions. It demonstrates that the geometry is notably tractable: for GTS, CTS, and RDTS the Fisher information matrices are diagonal in natural tail-parameter coordinates and the -connections are explicit, yielding -flat manifolds. These results are then leveraged for practical statistical applications, including bias reduction via Jeffreys priors and construction of Komaki-style Bayesian predictive priors, using the common geometric structure across tempered stable processes. Overall, the framework generalizes previous work on CTS relative entropy to a broader class of tempered stable models and provides concrete tools for inference in heavy-tailed time-series contexts. This contributes a rigorous, operationally useful information-geometric toolkit for tempering stable processes in finance and related fields.

Abstract

We find the information geometry of tempered stable processes. Beginning with the derivation of -divergence between two tempered stable processes, we obtain the corresponding Fisher information matrices and the -connections on their statistical manifolds. Furthermore, we explore statistical applications of this geometric framework. Various tempered stable processes such as generalized tempered stable processes, classical tempered stable processes, and rapidly-decreasing tempered stable processes are presented as illustrative examples.

Paper Structure

This paper contains 12 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Tempered stable processes
  3. CTS processes and GTS processes
  4. RDTS processes
  5. Information geometry for tempered stable processes
  6. GTS processes
  7. CTS processes
  8. RDTS processes
  9. Statistical applications
  10. Bias reduction
  11. Bayesian predictive priors
  12. Conclusion

Key Result

Proposition 1

Let $(X_t, \mathbb{P})_{t\in[0,T]}$ and $(X_t, \mathbb{Q})_{t\in[0,T]}$ be CTS processes with parameters $(a, C, \lambda_+, \lambda_-, m)$ and $(\tilde{a}, \tilde{C}, \tilde{\lambda}_{+}, \tilde{\lambda}_{-}, \tilde{m})$, respectively. Suppose $\mathbb{P}$ and $\mathbb{Q}$ are equivalent measures sa

Theorems & Definitions (9)

  • Proposition 1: Kim and Lee (2007)
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof