On convergence of the Mayer problems arising in the theory of financial markets with transaction cost

Paper: arXiv 2605.11717

Authors: Yuri Kabanov, Artur Sidorenko

Abstract

The geometric approach to financial markets with proportional transaction cost prescribes to imbed a specific model (of stock market, of currency market etc.), usually given in a parametric form, into a natural framework defined by the two random processes, S and K. The first one, d-dimensional, models the price evolution of basic securities while the second one, cone-valued, describes the evolution of the solvency set. It happened that the fundamental questions – no-arbitrage criteria, hedging problems, portfolio optimization – can be studied in this general setting opening the door to set-valued techniques. In this note we explore, in such a general framework, the stochastic Mayer control problem, consisting in the maximization of the expected utility of the portfolio terminal wealth. We get results on continuity of the optimal value and the optimal control under price approximations in a general multi-asset framework described by the geometric formalism.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of stochastic Mayer control problems in markets with transaction costs using set-valued analysis and random cones. While mathematically dense and novel in its geometric approach to continuity under price approximations, it lacks empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze convergence of optimal value & control in financial markets with transaction costs] --> B[Inputs: Price process S, Solvency cone process K, & Utility function]
    B --> C[Framework: Embed market model into Geometric/Set-Valued Formalism]
    C --> D[Problem Formulation: Stochastic Mayer Control Problem for terminal wealth utility]
    D --> E[Methodology: Apply price approximations & set-valued analysis]
    E --> F[Outcomes: Continuity of optimal value & convergence of optimal controls]