On the dual problem of utility maximization in incomplete markets. (English) Zbl 1333.91050
Summary: In this paper, we study the dual problem of the expected utility maximization in incomplete markets with bounded random endowment. We start with the problem formulated in [J. Cvitanić et al., Finance Stoch. 5, No. 2, 259–272 (2001; Zbl 0993.91018)] and prove the following statement: in the Brownian framework, the countably additive part \(\hat{Q}^r\) of the dual optimizer \(\hat{Q} \in(L^\infty)^\ast\) obtained in [Cvitanić et al., loc. cit.] can be represented by the terminal value of a supermartingale deflator \(Y\) defined in [D. Kramkov and W. Schachermayer, Ann. Appl. Probab. 9, No. 3, 904–950 (1999; Zbl 0967.91017)], which is a local martingale.
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