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

TL;DR

This paper investigates the convergence properties of the Mayer control problem in financial markets with transaction costs, using a geometric framework involving stochastic processes for prices and solvency sets.

Contribution

It extends the analysis of the Mayer problem to a general multi-asset setting with transaction costs, establishing continuity results under price approximations.

Findings

Proves continuity of the optimal value in the Mayer control problem.

Establishes stability of optimal controls under price approximations.

Provides a geometric framework for analyzing financial market models with transaction costs.

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.