$q_x$-Bass Martingales: Optimality, Existence and Convergence

Paper: arXiv 2610.20920

Authors: Antonio Marini

Abstract

We introduce the class of $q_x$-Bass martingales, extending the notion of $q$-Bass martingales. These are martingale couplings with prescribed marginals whose transition kernels are closest to a given family of reference kernels $(q_x)_{x \in \mathbb R^d}$ in the sense of a suitable weak optimal transport problem. We show that $q_x$-Bass martingales are optimizers of this problem and establish a strong duality result. Building on this duality, we introduce an algorithm for their computation and, under compactness assumptions on the prescribed marginals, prove its convergence, thereby establishing existence. We then specialize to one-dimensional marginals and show that the algorithm reduces to a fixed-point iteration.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, introducing a new class of martingales and proving their optimality and existence through advanced mathematical techniques. While it presents an algorithm, there’s no empirical validation or backtesting, placing it firmly in the ‘Lab Rats’ quadrant. The novelty is high due to the generalization of q-Bass martingales.

Research Flowchart

  flowchart TD
    A[Research Goal: Introduce and analyze $q_x$-Bass martingales] --> B{Key Methodology: Weak Optimal Transport Problem};
    B -- Input --> C[Data/Inputs: Prescribed Marginals, Reference Kernels $(q_x)$];
    B -- Method --> D{Computational Process: Algorithm based on Strong Duality};
    D -- Specific Case --> E[Computational Process: Fixed-point Iteration for 1D Marginals];
    D --> F[Key Finding: $q_x$-Bass Martingales are Optimizers];
    F & E --> G[Key Finding: Existence & Convergence of Algorithm];