Paper: arXiv 2411.11408

Authors: Julio Backhoff, Edoardo Kimani Bellotto

Abstract

In continuous time, the laws of martingales tend to be singular to each other. Notably, N. Gantert introduced the concept of specific relative entropy between real-valued continuous martingales, defined as a scaling limit of finite-dimensional relative entropies, and showed that this quantity is non-trivial despite the aforementioned mutual singularity of martingale laws. Our main mathematical contribution is to extend this object, originally restricted to one-dimensional martingales, to multiple dimensions. Among other results, we establish that Gantert’s inequality, bounding the specific relative entropy with respect to Wiener measure from below by an explicit functional of the quadratic variation, essentially carries over to higher dimensions. We also prove that this lower bound is tight, in the sense that it is the convex lower semicontinuous envelope of the specific relative entropy. This is a novel result even in dimension one. Finally we establish closed-form expressions for the specific relative entropy in simple multidimensional examples.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical extension of Gantert’s specific relative entropy to multidimensional continuous martingales, establishing tightness and closed-form expressions. It is highly sophisticated mathematically but purely theoretical without empirical backtesting or applied dataset analysis.

Research Flowchart

  flowchart TD
    A["Research Goal: Extend Specific Relative Entropy (SRE) of continuous martingales to multidimensional settings"] --> B["Inputs: Multidimensional Continuous Martingales & Wiener Measure"]
    
    B --> C["Methodology: Define SRE via scaling limits of finite-dimensional relative entropies"]
    
    C --> D["Analysis: Compute lower bounds using Quadratic Variation functionals & evaluate convex LSC envelopes"]
    
    D --> E["Finding 1: Proved Gantert's inequality extends to higher dimensions"]
    D --> F["Finding 2: Established tightness of lower bound via convex LSC envelope (novel for d=1)"]
    D --> G["Finding 3: Derived closed-form SRE expressions for multi-asset examples"]