Paper: arXiv 2312.15119

Authors: Andreas Søjmark, Fabrice Wunderlich

Abstract

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) Lévy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod’s J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework using Functional Central Limit Theorems (FCLTs), Skorokhod topologies (J1 and M1), and subordinated Lévy processes. While it provides a strong, unifying theoretical synthesis across mathematical finance and physics, it is purely theoretical with minimal empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Unified Functional CLT for CTRWs to Subordinated Lévy Noise] --> B[Input: Continuous-Time Random Walks & Skorokhod Spaces J1 / M1]
    B --> C[Methodology: Apply Weak Convergence of Stochastic Integrals]
    C --> D[Analyze J1 Topology: General Principles & Standard CTRW Convergence]
    C --> E[Analyze M1 Topology: Complex Dynamics & Unmatched Jump Limits]
    D --> F[Key Findings: Streamlined Framework across Physics, Finance & Econometrics]
    E --> F

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0 of 80 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.