Paper: arXiv 2609.08605

Authors: Jan Vecer

Abstract

Let (P) be a fixed physical law and let (Q) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps (Q) to (T_LQ), where (d(T_LQ)=L,dQ) and (L) is the terminal likelihood ratio. The forward relative-entropy projection minimizing (D_{\mathrm{KL}}(P\Vert Q)) commutes with this transform because its objective changes only by the constant (-E_P\log L). The minimal entropy martingale measure (MEMM) orientation (D_{\mathrm{KL}}(Q\Vert P)) does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the (Q)-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex (f)-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density (dP/dQ^*) is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

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, focusing on advanced mathematical proofs and derivations related to numeraire invariance and entropy measures. While it introduces novel insights into the consistency of pricing functionals under different entropy orientations, it lacks empirical validation or backtesting. The mathematical rigor is exceptional, but the absence of real-world data application places it firmly in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Numeraire Invariance of Entropy-Projected Martingale Measures] --> B{Methodology: Compare Forward vs. Backward Relative Entropy Projections};
    B --> C[Inputs: Physical Law P, Equivalent Martingale Measure Q, Numeraire L];
    C --> D{Computational Processes: Numeraire Change (T_LQ), Forward Entropy D_KL(P||Q), Backward Entropy D_KL(Q||P), Trinomial Counterexample};
    D --> E[Key Findings: Forward Entropy Commutes with Numeraire Change; Backward Entropy Does Not];
    E --> F[Outcomes: Forward selectors price consistently; Backward MEMM prices inconsistently (counterexample); Logarithmic f-divergence for uniform invariance; Forward projection density is log-optimal wealth];