Reciprocal Specific Relative Entropy between Continuous Martingales

Abstract: We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets Aldous (2013)). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.

• The paper introduces RSRE to quantify divergence between martingale laws and the Wiener measure by leveraging the quadratic variation process.
• It rigorously proves that RSRE uniquely selects the scaled neutral Wright-Fisher diffusion as the optimal solution in martingale optimal transport problems.
• The study bridges discrete trinomial approximations and continuous stochastic models, enhancing model selection and calibration in prediction markets.

Summary of "Reciprocal Specific Relative Entropy between Continuous Martingales" (2602.14776)

Introduction and Motivation

The paper introduces and studies a new notion of divergence between continuous martingales: the reciprocal specific relative entropy (RSRE). This functional quantifies the distance between a martingale law $\mathbb{Q}$ and the Wiener measure $\mathbb{W}$ via the quadratic variation process. Specifically, RSRE takes the form:

$$\mathfrak{h}(\mathbb{Q} \| \mathbb{W}) = \frac{1}{2}\mathbb{E}_{\mathbb{Q}}\left[\int_0^1 \Sigma_t\log(\Sigma_t) + 1 - \Sigma_t \, dt\right]$$

where $\Sigma_t = d\langle X\rangle_t / dt$ is the instantaneous quadratic variation under $\mathbb{Q}$. The divergence thus measures how much the volatility profile of the martingale deviates from that of Brownian motion.

Three main motivations are delineated:

1. RSRE serves as a unique selection criterion in divergence-based martingale optimization, particularly resolving the non-uniqueness of optimizers in the critical case of specific $p$-Wasserstein divergence as $p \to 2$.
2. RSRE emerges naturally as a continuum limit of relative entropy in discrete martingale models, especially those related to trinomial trees.
3. RSRE enjoys a reciprocal relationship with the traditional specific relative entropy, effectively swapping the roles of measures in the entropy calculation via a time-change argument.

Theoretical Development

Relationship with Specific $p$-Wasserstein Divergence

The paper rigorously establishes that RSRE arises as the gradient of the specific $p$-Wasserstein divergence at $p=2$. For martingale optimal transport problems with prescribed initial and terminal distributions, this derivative singles out a unique optimizer among the previously degenerate solutions. This is formally expressed as:

$\mathbb{W}$0

where $\mathbb{W}$1 is the class of win-martingales starting at $\mathbb{W}$2, terminating almost surely at $\mathbb{W}$3.

Limit Results for Trinomial Models

The RSRE appears as the limiting relative entropy between discrete martingale approximations (trinomial trees) as the mesh goes to zero. This provides continuity between discrete and continuous-time stochastic frameworks, reinforcing the naturality of RSRE as an entropic measure for martingale path laws.

Reciprocal Structure

The paper formally proves the reciprocal relationship between RSRE and the conventional specific relative entropy, leveraging time-change techniques and measure-theoretic results under stochastic differential equations.

Optimization Problem for Win-Martingales

The major technical contribution is the complete solution of an RSRE minimization problem over the set of win-martingales—martingales modeling prediction markets that terminate at binary values. Specifically, the minimization is over:

$\mathbb{W}$4

The authors derive the associated HJB equation, analyze the stochastic control structure, and establish properties of the quadratic variation. The unique optimizer is shown to be the scaled neutral Wright-Fisher diffusion:

$\mathbb{W}$5

This diffusion is traditionally used in population genetics to model gene frequency dynamics under neutrality, but here it arises as the unique martingale minimizing RSRE among all win-martingales.

The value function for this optimization admits an explicit formula and is characterized as the unique viscosity solution to the derived HJB equation.

Notably, the paper demonstrates that the quadratic variation process $\mathbb{W}$6 for this diffusion is itself a martingale on $\mathbb{W}$7—a technical property not previously documented in the literature.

Strong Claims and Numerical Results

• The RSRE minimization problem over win-martingales has a unique solution: the scaled neutral Wright-Fisher diffusion.
• For the specific $\mathbb{W}$8-Wasserstein divergence, every feasible martingale is optimal at $\mathbb{W}$9, but RSRE selects one uniquely.
• The quadratic variation process in the Wright-Fisher diffusion is itself a martingale.

Numerical simulations and density estimates solidify integrability assumptions required for the rigorous connection between RSRE and Wasserstein divergence.

Multidimensional Considerations

The paper speculates on the extension of RSRE to multidimensional martingales. While quantum (von Neumann) relative entropy between matrix-valued measures is proposed as a natural candidate, numerical evidence indicates that the multidimensional neutral Wright-Fisher diffusion does not minimize the associated RSRE, raising open questions for further research.

Implications and Future Directions

Practically, the RSRE provides a theoretically grounded divergence measure for model selection and calibration in stochastic models, with direct applications to prediction markets and mathematical finance. The unique identification of the Wright-Fisher diffusion as the optimal win-martingale suggests profound connections between population genetics processes and risk-neutral modeling.

Theoretically, RSRE resolves degeneracies in martingale optimal transport, opens new pathways for entropic regularization, and connects discrete and continuous stochastic frameworks. The reciprocal structure invites further exploration of measure-theoretic dualities in stochastic process divergence.

Future research may address:

• Extension to higher-dimensional martingale laws and identification of optimal processes under RSRE.
• Detailed study of quantum entropy divergences for matrix-valued measures.
• Applications to robust model calibration and uncertainty quantification in finance and other domains.

Conclusion

This paper rigorously defines and analyzes a new martingale divergence, the reciprocal specific relative entropy, demonstrating its role as a tie-breaking optimizer in specific Wasserstein divergence minimization and its emergence as the unique selection criterion in prediction market modeling via win-martingales. The scaled neutral Wright-Fisher diffusion is identified as the unique minimizer, with explicit solution characterization and strong technical properties. The work establishes deep links between entropic divergence, martingale optimal transport, and diffusion process modeling, laying a foundation for broad theoretical and practical developments in stochastic analysis and model selection (2602.14776).