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MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN INCOMPLETE MARKET
Author(s) -
Xia Jianming,
Yan JiaAn
Publication year - 2006
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.2006.00268.x
Subject(s) - semimartingale , martingale (probability theory) , mathematics , local martingale , duality (order theory) , portfolio optimization , mathematical economics , incomplete markets , probability measure , mathematical finance , lévy process , portfolio , economics , pure mathematics , discrete mathematics , finance , neoclassical economics
In this paper, for a process S , we establish a duality relation between K p , 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 K p 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.