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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.

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