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October 1997 On $L^2$-projections on a space of stochastic integrals
Thorsten Rheinländer, Martin Schweizer
Ann. Probab. 25(4): 1810-1831 (October 1997). DOI: 10.1214/aop/1023481112


Let $X$ be an $\mathbb{R}^d$-valued continuous semimartingale, $T$ a fixed time horizon and $\Theta$ the space of all $\mathbb{R}^d$ -valued predictable $X$ -integrable processes such that the stochastic integral $G(\vartheta)=\int\vartheta dX$ is a square-integrable semimartingale. A recent paper gives necessary and sufficient conditions on $X$ for $G_T(\Theta)$ to be closed in $L^2(P)$. In this paper, we describe the structure of the $L^2$-projection mapping an $\mathscr{F}_T$-measurable random variable $H \in L^2(P)$ on $G_T(\theta)$ and provide the resulting integrand $\vartheta^H \in \Theta$ feedback form. This is related to variance-optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on $X$. Our proofs use the variance-optimal martingale measure $\tilda{P}$ for $X$ and weighted norm inequalities relating $\tilda{P}$ to the original measure $P$.


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Thorsten Rheinländer. Martin Schweizer. "On $L^2$-projections on a space of stochastic integrals." Ann. Probab. 25 (4) 1810 - 1831, October 1997.


Published: October 1997
First available in Project Euclid: 7 June 2002

zbMATH: 0895.60051
MathSciNet: MR1487437
Digital Object Identifier: 10.1214/aop/1023481112

Primary: 60G48 , 60H05 , 90A09

Keywords: $ L^2$-projection , Kunita–Watanabe decomposition , Semimartingales , stochastic integrals , variance-optimal martingale measure , weighted norm inequalities

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 4 • October 1997
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