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**Cone-constrained continuous-time Markowitz problems.**
*(English)*
Zbl 1268.91162

The mean-variance portfolio selection problem (Markowitz problem) consists in finding in a financial market a self-financing strategy whose final wealth has maximal mean and minimal variance. In the present paper this problem is studied in continuous time in a general semimartingale model and under some constraints, meaning that each allowed trading strategy is restricted to always lie in a closed cone which might depend on the state and time in a predictable way. As in the unconstrained case, the solution to the Markowitz problem can be obtained by solving the particular mean-variance hedging problem of approximating in \(L^2\) a constant payoff by the terminal gains of a self-financing trading strategy. To get existence of a solution to the latter problem, it is shown first that the space of constrained terminal gains is closed in \(L^2\); this is sufficient if the constraints are in addition convex. The main focus and achievement of the paper is the subsequent structural description of the optimal strategy by its local properties. This is made possible by treating the approximation in \(L^2\) as a problem in stochastic optimal control. By exploiting the quadratic and conic structure of the problem, the authors obtain a decomposition of its value process into a sum with two opportunity processes \(L^\pm\) appearing as coefficients. Using the martingale optimality principle allows them to describe the drift of \(L^\pm\) and from there the optimal strategy locally in feedback form via the pointwise minimizers of two predictable functions. So, the martingale optimality principle translates into a drift condition for the semimartingale characteristics of \(L^\pm\) or equivalently into a coupled system of backward stochastic differential equations for \(L^\pm\). Verification results are proved as well. This explains and generalizes all results so far in the literature on the Markowitz problem under cone constraints. The generality of the framework allows us to capture a new behavior of the optimal strategy.

Reviewer: Yuliya S. Mishura (Kyïv)

### MSC:

91G10 | Portfolio theory |

91G80 | Financial applications of other theories |

93E20 | Optimal stochastic control |

60G48 | Generalizations of martingales |

49N10 | Linear-quadratic optimal control problems |

### Keywords:

Markowitz problem; cone constraints; portfolio selection; mean-variance hedging; stochastic control; semimartingales; backward stochastic differential equations; martingale optimality principle; opportunity process; linear-quadratic control; \(\varepsilon\)-martingales
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\textit{C. Czichowsky} and \textit{M. Schweizer}, Ann. Appl. Probab. 23, No. 2, 764--810 (2013; Zbl 1268.91162)

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