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In this paper, for a process S, we establish a duality relation between Kp, the ‐ closure of the space of claims in , which are attainable by “simple” strategies, and , all signed martingale measures with , where p≥ 1, q≥ 1 and . If there exists a with a.s., then Kp consists precisely of the random variables such that ϑ is predictable S‐integrable and for all . The duality relation corresponding to the case p=q= 2 is used to investigate the Markowitz's problem of mean–variance portfolio optimization in an incomplete market of semimartingale model via martingale/convex duality method. The duality relationship between the mean–variance efficient portfolios and the variance‐optimal signed martingale measure (VSMM) is established. It turns out that the so‐called market price of risk is just the standard deviation of the VSMM. An illustrative example of application to a geometric Lévy processes model is also given.
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