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**1 - 2**of**2**### Working Paper Series Cone-Constrained Continuous-Time Markowitz Problems Cone-Constrained Continuous-Time Markowitz Problems

"... Abstract The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in ..."

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Abstract The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and timedependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L 2 . Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L ± appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L ± or equivalently into a coupled system of backward stochastic differential equations for L ± . We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.

### Working Paper Series Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints out within the NCCR FINRISK project on "Mathematical Methods in Financial Risk Management" Convex duality in mean-variance hedging under convex trading constrain

"... Abstract 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 predictabl ..."

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Abstract 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. MSC 2010 Subject Classification: 60G48, 91G10, 93E20, 49N10, 60H05