A minimum variance result in continuous trading portfolio optimization
Management Science
Portfolio selection with transaction costs
Mathematics of Operations Research
European option pricing with transaction costs
SIAM Journal on Control and Optimization
Investment strategies under transaction costs: the finite horizon case
Management Science
Dynamic Asset Allocation in a Mean-Variance Framework
Management Science
Mean-Variance Portfolio Selection with Random Parameters in a Complete Market
Mathematics of Operations Research
Mathematics of Operations Research
Finite Horizon Optimal Investment and Consumption with Transaction Costs
SIAM Journal on Control and Optimization
Trend Following Trading under a Regime Switching Model
SIAM Journal on Financial Mathematics
Online portfolio selection: A survey
ACM Computing Surveys (CSUR)
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A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem, inherently with a finite time horizon. Via a series of transformations, the problem is turned into a so-called double obstacle problem, a well-studied problem in physics and PDE literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.