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

Abstract The geometric approach to multi-asset financial markets with proportional transaction cost prescribes embedding a specific model (of a stock market, of a 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 turned out that the fundamental questions (no-arbitrage criteria, hedging problems, portfolio optimization) can be studied in this general setting. 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 terminal wealth of the portfolio. We prove that the optimal value is continuous under price approximations. We also prove that the value of the limit model is attained by a strong strategy, and that an asymptotically optimal sequence of strategies has a subsequence converging in law to a compatible strategy.