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The variance-optimal martingale measure for continuous processes. (English) Zbl 0849.60042

Bernoulli 2, No. 1, 81-105 (1996); corrections ibid. 2, No. 4, 379-480 (1996).
Let \(S\) be a multidimensional semimartingale on \((\Omega, {\mathcal F}, P)\) and consider the set \({\mathcal M}^s (P)\) of all signed measures \(Q\) with \(P\)-square-integrable density and such that each elementary stochastic integral of \(X\) has \(Q\)-expectation 0. These measures come up in applications in financial mathematics. \(Q^{\text{opt}} \in {\mathcal M}^s (P)\) is called variance-optimal if its density has minimal \({\mathcal L}^2 (P)\)-norm. The main result is that \(Q^{\text{opt}}\) is automatically equivalent to \(P\) if \(S\) is continuous and if \({\mathcal M}^s (P)\) contains at least one element which is equivalent to \(P\). [Note that \({\mathcal M}^s\) should read \({\mathcal M}^e\) in Theorem 1.3.] For discontinuous \(S\), this conclusion fails in general. An approximation problem is discussed as an application.

MSC:

60G48 Generalizations of martingales
91B28 Finance etc. (MSC2000)
60H05 Stochastic integrals
60G35 Signal detection and filtering (aspects of stochastic processes)
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