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A unified characterization of \(q\)-optimal and minimal entropy martingale measures by semimartingale backward equations. (English) Zbl 1040.60058
Given an incomplete financial market model whose asset prices are described by continuous semimartingales, the authors develop a unified characterization of \(q\)-optimal martingale measures for \(q\in[0,+\infty )\). The \(q\)-optimal martingale measure, introduced by P. Grandits and L. Krawczyk [in: Séminaire de probabilités XXXII. Lect. Notes Math. 1686, 73–85 (1998; Zbl 0914.60017)], is a measure with minimal \(L^q\)-norm among all signed martingale measures. The authors express the densities of \(q\)-optimal martingale measures in terms of a solution of an associated backward semimartingale equation, a kind of backward stochastic differential equations introduced by R. J. Chitashvili [in: Probability theory and mathematical statistics. Lect. Notes Math. 1021, 73–92 (1983; Zbl 0543.93071)]. According to their characterization the variance-optimal martingale measure [introduced by M. Schweizer, Ann. Appl. Probab. 2, 171–179 (1992; Zbl 0742.60042)], the minimal entropy martingale measure [cf. P. Grandits and T. Rheinländer, Ann. Probab. 30, 1003–1038 (2002; Zbl 1049.60035)] and the minimal martingale measure appear as the special case \(q=2\), \(q=1\) and \(q=0\), respectively. Moreover, under the assumption of the reversed Hölder condition, the continuity in \(L^1\) and in entropy of the densities of \(q\)-optimal martingale measures with respect to \(q\) is shown with the help of the continuity properties of the solution of the associated backward semimartingale equation.

60H30 Applications of stochastic analysis (to PDEs, etc.)
91B28 Finance etc. (MSC2000)
90C39 Dynamic programming
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