Jilong Xu | Searcharxiv

arXiv2026

Preimage Regions of Symmetric Separable Maps on the Simplex: Convexity and Barycentric Star-Shapedness

We study preimage regions on the open probability simplex associated with symmetric separable functionally generated maps. The problem is a finite-dimensional geometric question about convexity and barycentric star-shapedness of these regions. In the portfolio interpretation, the regions consist of the points whose generated portfolio has no negative coordinate. For symmetric separable generators, the defining first-order inequalities split into a coordinate term and a symmetric aggregation term. This coordinate--aggregation decomposition is the main organizing device of the paper. We show that the aggregation term may destroy convexity, and may even destroy barycentric star-shapedness. In particular, moving closer to the barycenter need not preserve the long-only property. We then give a necessary and sufficient threshold criterion for barycentric star-shapedness and derive sufficient conditions that recover it. These conditions are expressed in terms of concavity and second-derivative domination for the aggregation function. The entropy case is the affine aggregation case, in which the long-only constraints reduce to coordinate thresholds.

math.MG↗

arXiv2026

The Geometry of Admissible Short Selling in Discrete-Time Stochastic Portfolio Theory

While discrete-time Stochastic Portfolio Theory (SPT) provides a robust framework for market analysis, existing work on functional generation has predominantly focused on long-only portfolios defined on the entire unit simplex. This paper extends the geometric framework of functional generation to the broader class of bankruptcy-proof long-short portfolios defined on local market state spaces. We establish that, within this admissible setting, pseudo-arbitrage is fully characterized by the concavity of the generating function on the market state space, thereby relaxing the usual global domain requirement. A central contribution of this work is a geometric characterization of the short-selling mechanism. We prove that the presence of short selling is equivalent to the negativity of the maximal concave extension of the generating potential. This phenomenon is linked to the steepness of the logarithmic gradient as the market approaches a zero boundary nested inside the simplex. To systematically exploit this mechanism, we introduce the barycentric scaling transformation, a constructive methodology that maps classical long-only generating functions onto restricted domains to engineer admissible strategies with controlled short-selling exposure. Finally, through the analysis of specific shrunken portfolios, we identify a geometric phase transition: under suitable boundary conditions, admissible strategies exhibit a long-only core and a short-selling region in a qualitative sense (without asserting an exact partition of the state space). This provides a unified geometric perspective on relative arbitrage beyond the long-only constraint.

math.OC↗