The Annals of Probability

Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims

Pascale Monat and Christophe Stricker

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Abstract

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.

Article information

Source
Ann. Probab., Volume 23, Number 2 (1995), 605-628.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988281

Digital Object Identifier
doi:10.1214/aop/1176988281

Mathematical Reviews number (MathSciNet)
MR1334163

Zentralblatt MATH identifier
0830.60040

JSTOR
links.jstor.org

Subjects
Primary: 60G48: Generalizations of martingales
Secondary: 60H05: Stochastic integrals 90A09

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

Citation

Monat, Pascale; Stricker, Christophe. Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims. Ann. Probab. 23 (1995), no. 2, 605--628. doi:10.1214/aop/1176988281. https://projecteuclid.org/euclid.aop/1176988281


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