Strictly monotone mean-variance preferences with applications to portfolio selection

This paper extends the monotone mean-variance (MMV) preference to a broader class of strictly monotone mean-variance (SMMV) preferences, and demonstrates its applications to portfolio selection problems. For the single-period portfolio problem under the SMMV preference, we derive the gradient condition for the optimal strategy, and investigate its association with the optimal mean-variance (MV) static strategy. A novel contribution of this work is the reduction of the problem to solving a set of linear equations by analyzing the saddle point of some minimax problem. Building on this advancement, we conduct numerical experiments and compare our results with those of Maccheroni, et al. (Math. Finance 19(3): 487-521, 2009). The findings indicate that our SMMV preferences provide a more rational basis for assessing given prospects. For the continuous-time portfolio problem with the SMMV preference, we consider continuous price processes with random coefficients. We establish the condition under which the optimal dynamic strategies for SMMV and MV preferences coincide, and characterize the optimal solution using the dynamic programming principle and the martingale convex duality method, respectively. Consequently, the problem is reduced to solving a stochastic Hamilton-Jacobi-Bellman-Isaacs equation, or a multi-stage linear-quadratic optimization problem with the embedding technique.