Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market.

*(English)*Zbl 1082.91050Summary: This paper concerns the problems of quadratic hedging and pricing, and mean-variance portfolio selection in an incomplete market setting with continuous trading, multiple assets, and Brownian information. In particular, we assume throughout that the parameters describing the market model may be random processes. We approach these problems from the perspective of linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs); that is, we focus on the so-called stochastic Riccati equation (SRE) associated with the problem. Excepting certain special cases, solvability of the SRE remains an open question. Our primary theoretical contribution is a proof of existence and uniqueness of solutions of the SRE associated with the quadratic hedging and mean-variance problems. In addition, we derive closed-form expressions for the optimal portfolios and efficient frontier in terms of the solution of the SRE. A generalization of the Mutual Fund Theorem is also obtained.

##### MSC:

91G10 | Portfolio theory |

49N10 | Linear-quadratic optimal control problems |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

93E20 | Optimal stochastic control |