Solving Of Distributionally Robust Portfolio Optimization Problem Based On Burg Entropy-Divergence

Portfolio optimization models are often used to provide decision-making and measure investment risks for investors in the fields of finance,bonds and stocks.However,the rate of return on investment varies with various factors,which make us to get some information about it based on historical data rather than its specific distribution.So the problem of portfolio optimization with uncertain distribution has great research value.This paper mainly considers the following optimization models Where r is the investment return function,the random vector ??Rk is the return on investment,The decision variable x?X represents the portfolio selection of investment,and X(?)Rd is a non-empty compact set.The confidence level ? and ? represent the probability distribution on the bounded support set.P is the probability distribution on the bounded support set(?).We call the portfolio optimization problem with uncertain distribution as the distributed robust portfolio optimization model.The main idea in this paper is to apply BE-divergence theory,measure transformation and duality theory to transform the portfolio optimization problem with uncertain distribution into the portfolio optimization problem with empirical distribution on the basis of distributionally robust optimization.Specifically,the uncertainty set of distribution firstly is constructed according to the BE-divergence distance between the empirical distribution po obtained from historical data and the unknown distribution P.The portfolio optimization problem with unknown distribution P is transformed into a convex optimization problem with likelihood ratio l(?)under empirical distribution p0 by measure transformation.Finally,the equivalent portfolio optimization problem is obtained by equivalent transformation using convex optimization theory.For the equivalence problem,we use the Delta normal method to get the value under a certain confidence level and continue to solve it with the method of sample mean.It is concluded that the distributed robust portfolio optimization model is solvable based on BE-divergence.In the first part of this paper,the research status of portfolio optimization,the theory of distributed robust optimization and its basic concepts are introduced.In the second part,the equivalent form of the original problem is solved based on divergence theory,measure transformation and convex optimization theory.In the third part,the equivalence problem is further solved.Firstly,we get the VaR value at the confidence level by Delta normal method.Secondly,we use the sample mean method to solve the problem and verify its convergence.In the fourth part of the paper,a example is given to illustrate the application.