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Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

Part of:Miscellaneous topics in calculus of variations and optimal controlStochastic systems and controlStochastic analysisMathematical financeStochastic processes

Published online by Cambridge University Press:  04 January 2016

Christoph Czichowsky and
Martin Schweizer
Christoph Czichowsky*
Affiliation:
University of Vienna
Martin Schweizer*
Affiliation:
ETH Zürich and Swiss Finance Institute
*
∗ Postal address: Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A–1090 Vienna, Austria. Email address: christoph.czichowsky@univie.ac.at
∗∗ Postal address: Department of Mathematics, ETH Zürich, Rämistrasse 101, CH–8092, Zürich, Switzerland. Email address: martin.schweizer@math.ethz.ch
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Abstract

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We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L 2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

Keywords

Mean-variance hedgingconstraintsstochastic integralconvex duality

MSC classification

Primary:60G48: Generalizations of martingales91G10: Portfolio theory93E20: Optimal stochastic control49N10: Linear-quadratic problems
Secondary:60H05: Stochastic integrals

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012, pp. 1084 - 1112
Copyright
© Applied Probability Trust 

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