Paper: arXiv 2403.13791

Authors: Tahir Choulli, Martin Schweizer

Abstract

The classic stochastic Fubini theorem says that if one stochastically integrates with respect to a semimartingale $S$ an $η(dz)$-mixture of $z$-parametrized integrands $ψ^z$, the result is just the $η(dz)$-mixture of the individual $z$-parametrized stochastic integrals $\intψ^z{d}S.$ But if one wants to use such a result for the study of Volterra semimartingales of the form $ X_t =\int_0^t Ψ_{t,s}dS_s, t \geq0,$ the classic assumption that one has a fixed measure $η$ is too restrictive; the mixture over the integrands needs to be taken instead with respect to a stochastic kernel on the parameter space. To handle that situation and prove a corresponding new stochastic Fubini theorem, we introduce a new notion of measure-valued stochastic integration with respect to a general multidimensional semimartingale. As an application, we show how this allows to handle a class of quite general stochastic Volterra semimartingales.

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 highly advanced measure-theoretic probability theory and stochastic calculus to extend stochastic Fubini theorems for Volterra semimartingales. While highly rigorous and novel from a pure mathematical finance perspective, it lacks empirical data, backtesting, or code implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Stochastic Fubini Theorem to Volterra Semimartingales] --> B[Input: Multidimensional Semimartingale S & Stochastic Kernels]
    B --> C[Identify Limitation: Classic fixed measure η is too restrictive for X_t = ∫ Ψ_t,s dS_s]
    C --> D[Methodology: Develop novel Measure-Valued Stochastic Integration theory]
    D --> E[Prove New Stochastic Fubini Theorem using Stochastic Kernels]
    E --> F[Outcome: General Framework for Analyzing Volterra Semimartingales & Cross-Asset Applications]

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0 of 31 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.