The Analysis Of Optimal Dynamic Portfolio Policy

This paper is a survey on optimal dynamic portfolio selection.The process of optimal dynamic portfolio selection is to seek the benificial selection among wealth to achieve the maximization of expected value of terminal wealth and the minimization of the variance of the terminal wealth. This article introduce the most general multiperiod mean-variance formulation first,then discuss some complicated phenomenons which are relative with the real situation gradually.From the very beginning,we introduce Markowitz's the formulation of multiperiod mean-variance formulation. Min var(xT)An equivalent formulation to either or isThe optimal multiperiod portfolio policy for above problem is stated as follow:Then,Roy's safety-first approach is extended to multi-period selection problems.A op-timal multi-period portfolio policy is sought to minimize the probability that the terminal wealth is below a preselected level. The selection of a optimal multi-period portfolio can be stated as follow:The analytical solution is achieved for this problemNext,we describe Duan Li's the continuous-time mean-variance portfolio selection.The aim of this model is to maximize the expected terminal return and minimize the variance of the terminal wealth.By putting weights on the two criteria one obtains a single objective stochastic control problem.The optimal dynamic portfolio selection mean-variance problem (1)is specified by the following form:Min (J1(u(.)), J2(u(.)))= (-Ex(T), Var x(T))s.t. M(.)∈LF2(0, T;Rm), and (x(.), u(.))satisfyWe obtain the solution of original problem (2) by dealing with the portfolio control problem as follow.Minimize J1 (u(.))+μJ2(u(.))=-Ex(T)+μVarx(T) s.t. u(.)∈L2F(0, T;Rm), and (x(.), u(.))satisfyThe solution isThe portfolio policy selection of problem (2) is given by (4) withγ=γ=λ/2μandλgiven by (3).Then we get through a important portfolio policy selection models. It is the Shuxiang Xie's continuous-time mean-variance portfolio selection with liability and regime switch-ing.In this model,we assume that the risky stock's price is governed by a Markovian regime-switching geometric Brownian motion,and the liability follows a Markovian regime-switching Brownian motion with drift,respectively.Now,the mean-variance optimization problem with exogenous liability in a Markov- modulated model can be formulated asWe give the efficient portfolio and mean-variance efficient frontier for original problem (P1).Theorem 1 Suppose E∫0T b2(t, M(t))dt> 0, we have Moreover, the efficient portfolio corresponding to z, the wealth levelX(t)and market modeM(t),i where, among all the wealth process X(·) satisfy E[X(T)]= z,isVar[X(T)]= Further,the expressiondisclose the minimum variance,namely,the minimum possible vari-ance achievable by an admissble portfolio,along with the portfolio that attains this minimum variance. Theorem2 The minimum terminal variance iswith the corresponding expected terminal wealth and the corresponding Lagrange multiplierλmin*= 0. Moreover,the portfolio that achieves the above minimum variance,is...