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

2026-05-12

AI Summary

This study investigates the continuity of the optimal value and optimal strategies for a stochastic Mayer control problem in a multi-asset market with proportional transaction costs under price perturbations. Within a geometric financial modeling framework, the market is embedded into a set-valued structure defined by the price process and the solvency cone process. Combining stochastic control theory with set-valued analysis, the paper systematically examines the maximization of expected terminal utility of wealth. It establishes, for the first time in a general multi-asset geometric setting, that both the optimal value function and the optimal control strategies converge continuously as the price processes converge sequentially. This result not only extends the applicability of set-valued analysis in financial control problems but also provides a rigorous theoretical foundation for numerical approximation and robust optimization.

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.

Problem

• Mayer problem
• transaction cost
• financial markets
• stochastic control
• portfolio optimization

Innovation

• geometric market model
• transaction costs
• stochastic Mayer problem
• set-valued analysis
• continuity of optimal control