Paper: arXiv 2411.05470

Authors: Henry Chiu

Abstract

We present a non-probabilistic, path-by-path framework for studying path-dependent (i.e., where weight is a functional of time and historical time-series), long-only portfolio allocation in continuous-time based on [Chiu & Cont ‘23], where the fundamental concept of self-financing was introduced, independent of any integration theory. In this article, we extend this concept to a portfolio allocation strategy and characterize it by a path-dependent partial differential equation. We derive the general explicit solution that describes the evolution of wealth in generic markets, including price paths that may not evolve continuously or exhibit variation of any order. Explicit solution examples are provided. As an application of our continuous-time, path-dependent framework, we extend an aggregating algorithm of [Vovk ‘90] and the universal algorithm of [Cover ‘91] to continuous-time algorithms that combine multiple strategies into a single strategy. These continuous-time (meta) algorithms take multiple strategies as input (which may themselves be generated by other algorithms) and track the wealth generated by the best individual strategy and the best convex combination of strategies, with tracking error bounds in log wealth of order O(1) and O(ln t), respectively. This work extends Cover’s theorem [Cover ‘91, Thm 6.1] to a continuous-time, model-free setting.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This purely theoretical paper provides rigorous mathematical extensions of Vovk’s aggregating algorithm and Cover’s universal portfolio to a continuous-time, model-free setting using path-dependent PDEs. While highly novel and mathematically sophisticated, it lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Develop model-free, continuous-time framework for path-dependent portfolio allocation"] --> B["Base Framework & Inputs: Non-probabilistic path-by-path model and price paths (Chiu & Cont '23)"]
    B --> C["Core Methodology: Extend self-financing concept to allocations & characterize via path-dependent PDE"]
    C --> D["General Solution: Derive explicit wealth evolution equation for generic, non-continuous markets"]
    D --> E["Continuous-Time Meta-Algorithms: Extend Vovk ('90) and Cover ('91) aggregating algorithms"]
    E --> F["Key Outcomes: Model-free Cover's Theorem extension tracking best strategies with O(1) and O(ln t) error bounds"]