Paper: arXiv 2312.01730

Authors: Weixuan Xia

Abstract

In this paper we study set-valued Volterra-type stochastic integrals driven by Lévy processes. Upon extending the classical definitions of set-valued stochastic integral functionals to convoluted integrals with square-integrable kernels, set-valued convoluted stochastic integrals are defined by taking the closed decomposable hull of the integral functionals for generic time. We show that, aside from well-established results for set-valued Itô integrals, while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are guaranteed to be integrably bounded for bounded integrands, this is not true when the random measure is of infinite variation. For indefinite integrals, we prove that it is a mutual effect of kernel singularity and jumps that the set-valued convoluted integrals are possibly explosive and take extended vector values. These results have some important implications on how set-valued fractional dynamical systems are to be constructed in general. Two classes of set-monotone processes are studied for practical interests in economic and financial modeling.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a dense, highly rigorous mathematical framework for set-valued stochastic integrals driven by Lévy processes with singular kernels. While highly sophisticated in pure probability theory, it is entirely theoretical with no empirical backtesting, datasets, or code provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Define set-valued Volterra stochastic integrals driven by Lévy processes] --> B[Input: Lévy processes, square-integrable kernels & set-valued integrands]
    B --> C[Methodology: Extend set-valued integral functionals to convoluted kernels]
    C --> D[Compute: Take closed decomposable hull of integral functionals for generic time]
    D --> E[Finding 1: Finite-variation Poisson measures yield integrably bounded integrals; infinite-variation fails]
    D --> F[Finding 2: Kernel singularity + jumps cause indefinite integrals to explode & take extended vector values]
    E --> G[Outcome: Framework for set-valued fractional dynamical systems & set-monotone financial models]
    F --> G