The Maximum Principle For One Kind Of Stochastic Optimization Problem And Application In Continuous-time Mean-variance Portfolio Selection With Bankruptcy Prohibition

A continuous-time mean-variance portfolio selection problem is studied where all the market coefficients are random and the wealth process under any admissible trading strategy is not allowed to be below zero at any time.We find that we can replace the pointwise constraint x(t) ≥ 0 by the terminal constraint x(T) ≥ 0. The problem is completely solved using Ekeland variational principle to get the maximum principle for one kind of stochastic optimization problem motivated by continuous-time mean-variance portfolio selection with bankruptcy prohibition.Then we find the variance minimizing portfolios are derived as the replicating portfolio of some contingent claims.and the variance minimizing frontier is obtained.Finally, the efficient frontier is identified as an appropriate portion of the variance minimizing frontier after the monotonicity of the minimum variance on the expected terminal wealth over this portion is proved and all the efficient portfolios are found. The continuous-time mean-variance portfolio selection with bankruptcy prohibition to the investor in financial market can be studied in. our framework where the wealth equation may have nonlinear coefficient.