The Annals of Probability

Approximating Random Variables by Stochastic Integrals

Martin Schweizer

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Let $X$ be a semimartingale and $\Theta$ the space of all predictable $X$-integrable processes $\vartheta$ such that $\int\vartheta dX$ is in the space $\mathscr{S}^2$ of semimartingales. We consider the problem of approximating a given random variable $H \in\mathscr{L}^2$ by a stochastic integral $\int^T_0 \vartheta_s dX_s$, with respect to the $\mathscr{L}^2$-norm. If $X$ is special and has the form $X = X_0 + M + \int \alpha d\langle M\rangle$, we construct a solution in feedback form under the assumptions that $\int \alpha^2 d\langle M\rangle$ is deterministic and that $H$ admits a strong F-S decomposition into a constant, a stochastic integral of $X$ and a martingale part orthogonal to $M$. We provide sufficient conditions for the existence of such a decomposition, and we give several applications to quadratic optimization problems arising in financial mathematics.

Article information

Ann. Probab., Volume 22, Number 3 (1994), 1536-1575.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Semimartingales stochastic integrals strong F-S decomposition mean-variance tradeoff option pricing financial mathematics


Schweizer, Martin. Approximating Random Variables by Stochastic Integrals. Ann. Probab. 22 (1994), no. 3, 1536--1575. doi:10.1214/aop/1176988611.

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