Theoretical Analysis And Numerical Algorithm On Two Kinds Of Portfolio Model With Stochastic Factors

The research on portfolio problem has been an important subject in finance,mathematics and statistics.In the past decade,to describe the impact of the changes in financial markets on investment strategies,a large number of scholars combine stochastic factors with portfolio models.For example,the stochastic slow factor is introduced to describe the long-term fluctuations of risk assets(such as stock disaster)on investment,and delay factors are introduced to describe the historical trend of risk assets.This paper studies some problems of the portfolio model with a slow factor(the slow factor model)and the portfolio model with delay factors(the delay factor model),including the parameter identification theory and numerical algorithms of the slow factor model,the calculation theory of the delay factor model and game problems.The theoretical analysis results and numerical algorithms obtained in this paper enrich the theory of portfolio optimization problems with stochastic factor.For the slow factor model,our research is divided into two parts:(1)Identify the constant parameters—expected rate of return and absolute risk aversion coefficient in the portfolio model with a slow factor.First,through the asymptotic expansion method,we obtain the approximate solution of the value function.Next,from the approximate solution and observation data of the value function,the parameters—expected rate of return and absolute risk aversion coefficient can be estimated by the least square method.Moreover,the uniqueness and stability of the estimators of the constant parameters are proved.Finally,a numerical example is used to show the effectiveness and reliability of the results in the inverse problem.(2)Identify the functional parameter—the expected rate of return which is a function of the slow factor.We assume that the observed data is affected by the random error with heteroscedasticity.First,we propose an online weighted regularization method to fit the value function and the expected rate of return can be obtained by the numerical integration.Then,theoretical analysis and numerical experiments are provided.Finally,we introduce the online weighted regularization method and provide the selection method of weighted coefficients and application cases in other fields.For delay factor model,our research is also divided into two parts:(1)Improve the method of solving the delay factor model and provide a new Hamilton-Jacobi-Bellman equation(HJB equation)of the value function.Since,in the portfolio model with the delay factors,the initial value of the wealth process Xt is a function,the model is an infinite dimensional problem and the classical stochastic dynamic programming principle can not be used directly.Regarding the delay factors as a whole,the original problem is gradually reduced to a finite dimensional problem.Then through the principle of stochastic dynamic programming,the HJB equation of the value function is obtained.Finally,the corresponding verification theorem and the numerical example are provided.(2)Research the game problems of the delay factor model.Firstly,we study the n-investor game problem.Each investor allocates capital between their own risky asset and risk-free asset,but they will compare their future earnings with others and change their investment strategy.We use the method in the first part of the delay factor model to reduce the problem into a finite dimensional problem and get the HJB equation of investors’ value function.Combing the definition of Nash equilibrium,the investment strategy of each investor is obtained.At the same time,the corresponding verification theorem and the numerical example are provided.Then,we analyze the mean-field game when investors tend to infinity.That is,all the external influences on investors are approximately considered as an external field.Finally,we analyze the mathematical relationship between n-investor game and mean-field game.The main contributions of this paper are as follows:(1)We identify constant parameters and functional parameters in slow factor model and provide the necessary theoretical proof and numerical examples.(2)We propose a new method for reconstructing one-dimensional function—on line weighted regularization method,and provides the selection method of weighted coefficients.(3)We improve the method of solving the delay factor model which can be reduced to a finite dimensional problem.(4)The finite population game problem and the mean-field game problem of the delay factor model are analyzed,and the Nash equilibrium of the two problems is provided.