Short communication: a note on utility maximization with proportional transaction costs and stability of optimal portfolios

The aim of this short note is to establish a limit theorem for the optimal trading strategies in the setup of the utility maximization problem with proportional transaction costs. This limit theorem resolves the open question from [E. Bayraktar, L. Dolinskyi, and Y. Dolinsky, Finance Stoch., 24 (2020), pp. 1013-1034]. The main idea of our proof is to establish a uniqueness result for the optimal strategy. The proof of the uniqueness is heavily based on the dual approach which was developed recently in [Ch. Czichowsky and W. Schachermayer, Ann. Appl. Probab., 26 (2016), pp. 1888- 1941; Ch. Czichowsky, W. Schachermayer, and J. Yang, Math. Finance, 27 (2017), pp. 623-658; Ch. Czichowsky et al., Finance Stoch., 22 (2018), pp. 161-180].