Continuous-time mean–variance portfolio selection with random horizon in an incomplete market

Abstract

In this paper, we consider a continuous-time mean–variance portfolio selection problem with random market parameters and random time horizon in an incomplete market. This problem will be formulated as a linearly constrained stochastic linear quadratic (LQ) optimal control problem. The solvability of this LQ problem will be reduced to the global solvability of two backward stochastic differential equations (BSDEs). One is conventionally called a stochastic Riccati equation (SRE), and the other is referred to as an auxiliary BSDE. We shall apply the martingales of bounded mean oscillation, briefly called BMO-martingales, to provide a direct and simplified proof of the solvability of the two BSDEs. We also derive closed-form expressions for both the optimal portfolios and the efficient frontier in terms of the solutions of the two BSDEs.