Paper: arXiv 2308.03858

Authors: Christa Cuchiero, Tonio Möllmann, Josef Teichmann

Abstract

Generalized Feller theory provides an important analog to Feller theory beyond locally compact state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with standard and generalized ones. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e. a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.

Complexity vs Empirical Score

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

Why this score: The paper presents deep, rigorous mathematical extensions of generalized Feller theory applicable to signature SDEs, Markovian lifts, and infinite-dimensional affine processes. However, as a purely theoretical probability paper, it contains no empirical backtesting or historical data analysis. Its value lies in providing rigorous proofs and foundational theory for advanced quantitative modeling.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Generalized Feller Theory beyond locally compact spaces] --> B[Inputs: SPDEs, Fractional Processes, & Signature SDEs]
    B --> C[Methodology: Apply Method of Characteristics to Transport Equations]
    C --> D[Analysis: Rigorous proofs of path properties & construction]
    D --> E[Counterexample Testing: Validate necessity of basic Feller definitions]
    E --> F[Outcomes: Extended Feller concept & generic framework for weighted spaces]