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.


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