The Problems Of Terminal Wealth And Mean Variance In A D-MAP Model

Not only is the price of risky assets random,the economic environment state and decisions of investors are also random in real financial market.The rapid change of economic environment makes most investors uncertain about their future economic behavior.Therefore,the batch Markov Arrival Process or Markov Arrival Process was introduced into the risk model by many scholars.The risk model proposed can not only describe the change of economic environment condition but?also reflect the variety of economic behavior of investors,which is more consistent with the real financial market in daily life.In order to give a more realistic risk model,we introduce the discrete-time Markov Arrival Process(D-MAP)to modulate the financial market(N period),and discuss the problems of terminal wealth and mean variance of discrete time.The batch Markov Arrival Process is a stochastic point process generated by standard Poisson process.It is a two-dimensional Markov chain,which contains a counting process and a phase process.In particular,it is called the D-MAP when the time is discrete and the elements of phase process contain only two states.The D-MAP is a special case of the BMAP,and the number of events arriving at discrete-time points is either zero or only one,so the D-MAP is a stochastic process containing two different state spaces.Under the modulation of the D-MAP,the investor has no opportunity or has opportunity to invest in risky assets on some discrete jumping points of D-MAP,and the investor be in different environment condition on different jumping point.This is good for describing the change of the economic environmental state and the economic behaviors of investors,which are closer to the real life.We mainly consider two aspects after introducing the D-MAP to modulate the financial market.On the one hand,choosing to maximize the expected utility of terminal wealth as the objective function,we consider an N-period financial market with one riskless asset(such as bond)and a kind of risky asset(such as stock).The economic environmental state is modulated by the markov chain and the investor's opportunity of risky asset is modulated.by the counting process in the D-MAP.We solve the optimization investment problem of terminal wealth of the general utility function and specific utility function,finding the optimal investment strategy by auxiliary markov modulated model and markov decision model.Taking power utility function as an example,the influence of investment opportunity and environment state on value function is analyzed.On the other hand,we choose to minimize risk(variance)as the objective function while fixing the expectation of the terminal wealth-the mean-variance criterion.First,the mean variance problem is converted to the Lagrange problem.By auxiliary markov modulated model and markov decision model,we solve an auxiliary problem of Lagrange problem and the optimal investment problem with the mean-variance criterion,finding the optimal solution of the investment strategy.Finally,we summarize and prospect the results of thispaper,offering the possible direction studied in the future.