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The Föllmer-Schweizer decomposition. (English) Zbl 0877.60029

Engelbert, Hans-Jürgen (ed.) et al., Stochastic processes and related topics. Proceedings of the 10th winter school, Siegmundsberg, Germany, March 13–19, 1994. Amsterdam: Gordon and Breach Publishers. Stochastics Monogr. 10, 77-89 (1996).
Assume that \(X=X_0+M+A\) is a vector-valued special semimartingale in its canonical representation and \(H\) is a random variable in \(L^2\). In the theory of mean-variance hedging of contingent claims of Föllmer and Schweizer it is important to know whether \(H\) can be written as \(H=H_0+\xi\cdot X+L\) where \(L\) is a martingale orthogonal to all integrals \(\theta\cdot M\). If \(X\) is a square-integrable martingale such a decomposition always exists: this is the classic result due to Kunita and Watanabe in the scalar case and Galtchouk in the vector case. For a general semimartingale, the decomposition may not exist. The paper gives a brief survey of results on existence. Further information can be found in [Delbaen et al., Finance and Stochastics 1, 181-227 (1997)].
For the entire collection see [Zbl 0851.00074].

MSC:

60G44 Martingales with continuous parameter
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