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April, 1995 Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims
Pascale Monat, Christophe Stricker
Ann. Probab. 23(2): 605-628 (April, 1995). DOI: 10.1214/aop/1176988281


Let $X$ be an $\mathbb{R}^d$-valued special semimartingale on a probability space $(\Omega, \mathscr{F}, (\mathscr{F}_t)_{0\leq t \leq T},P)$ with decomposition $X = X_0 + M + A$ and $\Theta$ the space of all predictable, $X$-integrable processes $\theta$ such that $\int\theta dX$ is in the space $\mathscr{J}^2$ of semimartingales. If $H$ is a random variable in $\mathscr{L}^2$, we prove, under additional assumptions on the process $X$, that $H$ can be written as the sum of an $\mathscr{F}_0$-measurable random variable $H_0$, a stochastic integral of $X$ and a martingale part orthogonal to $M$. Moreover, this decomposition is unique and the function mapping $H$ with its decomposition is continuous with respect to the $\mathscr{L}^2$-norm. Finally, we deduce from this continuity that the subspace of $\mathscr{L}^2$ generated by $\int\theta dX$, where $\theta\in \Theta$, is closed in $\mathscr{L}^2$, and we give some applications of this result to financial mathematics.


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Pascale Monat. Christophe Stricker. "Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims." Ann. Probab. 23 (2) 605 - 628, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0830.60040
MathSciNet: MR1334163
Digital Object Identifier: 10.1214/aop/1176988281

Primary: 60G48
Secondary: 60H05 , 90A09

Keywords: Follmer-Schweizer decomposition , Kunita-Watanabe decomposition , orthogonal martingales , Semimartingales , stochastic integrals

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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