摘要：

This thesis contains three scientific articles on the problem of utility maximization in a financial market with proportional transaction costs. One of the most revealing approaches to this problem consists in passing to the asymptotic analysis of the impact of small transaction costs, where the maximum expected utility is approximated locally by a Taylor series expansion. In recent works by Kallsen and Muhle-Karbe (2012, 2013) dealing with this issue, heuristic arguments are used for deriving a candidate trading strategy which is expected to be optimal to the leading order, along with asymptotic formulas for the associated trading volume and the effect of the small transaction costs. The subject of this thesis is a rigorous proof verifying the results of Kallsen and Muhle-Karbe with precise formulation of sufficient regularity assumptions on the market model. The study is carried out in three stages considering the exponential utility function, general utility functions and trading volume of the leading-order optimal strategy, respectively. The key idea throughout is the utilization of the so-called shadow price process which facilitates a transformation of the original problem with transaction costs into an equivalent frictionless one. The proof of the approximate optimality of the proposed strategy is based on the martingale method (duality theory), while the verification of the asymptotic formula for the respective trading volume and the impact of transaction costs relies on the ergodic property of reflected Brownian motion. The entire method works within a general one-dimensional Ito framework (i.e., the stock price is modelled by a general Ito process), which provides an access to stochastic volatility models.

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