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Risk minimization under transaction costs. (English) Zbl 1007.91021

This paper deals with the problem of minimizing the risk of the trading position at fixed date in a market with incomplete information, proportional transaction costs, and constraints on strategies. The author considers a market with a risky asset \(X\) and a riskless asset \(B\). \(X\) is a continuous semimartingale, the transaction cost up to time \(t\) is equal to \(C_{t}(\theta)= \int_{[0,t]}k_{s} d|D\theta_{s}|\), where a cost of \(k_{t}\) is incurred for each share traded, \(\theta_{t}\) is a number of shares invested in the risky asset at time \(t\), \(D\theta\) is the Radon measure. The considered risk functionals are general enough to allow the minimization of shortfall or the minimization of the coherent risk measure and the maximization of the expected utility. A theorem on the existence of optimal strategies in markets with incomplete information and transaction costs is proved. The constrained case is studied for two types of constraints: those on the position in the risky asset (such as limits on short-selling) and those on the portfolio value (such as margin requirements).

MSC:

91B28 Finance etc. (MSC2000)
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