Risk control over bankruptcy in dynamic portfolio selection: a generalized mean-variance formulation

For an investor to claim his wealth resulted from his multiperiod portfolio policy, he has to sustain a possibility of bankruptcy before reaching the end of an investment horizon. Risk control over bankruptcy is thus an indispensable ingredient of optimal dynamic portfolio selection. We propose in this note a generalized mean-variance model via which an optimal investment policy can be generated to help investors not only achieve an optimal return in the sense of a mean-variance tradeoff, but also have a good risk control over bankruptcy. One key difficulty in solving the proposed generalized mean-variance model is the nonseparability in the associated stochastic control problem in the sense of dynamic programming. A solution scheme using embedding is developed in this note to overcome this difficulty and to obtain an analytical optimal portfolio policy.