Since Magill and Constantinidesde study of the optimal consumption and investment problem, optimal consumption and investment problem has been a hot research subject for the financial economics. How to establish mathematical model to describe the financial market friction and what’s the investors’investment strategy in such case is a very challenging task. The effect of the Convex analysis methods in a variety of assets with transaction cost investment consumption model research is increasing. Y. Kabanov、C. Kluppelberg have studied stochastic optimal conditions under the restriction of infinite time range, and proved the utility function is defined by a Bellman function is the unique viscosity solution of HJB equation.Y. Kabanov and C. Kluppelberg’s study of optimal investment and consumption is not universal, we study in infinite time range, using optimal control theory and cone geometry consumption, the model is more practical significance. In particular to say, we assume the domain K is closed convex cone、combined value V is a solution of control process and linear equations dVi,t=Vi,t-dYi,t+dθi,t-dCi,t,Vi,0-=xi,i=1,2,…,d C=(ct) is interpreted as consumption process,θ=(θt) is described as the transfer of funds, and the model has m(m≥2) consumers. Morever, we hypothesis utility function u:W→R+is a concave function, which u(0)=0, u(x)/|x|→0,(when|x|→∞).Do the objective function that is the Bellman functionAnd prove J(x) is a unique viscosity solution of a HJB (Hamilton-Jacobi-Bellman)’s Dirchlet problems. So when the total reimbursement values of m consumers isx, through J(x) and dVi,t=Vi,tdYi,t,+dCi,t,we can get the optimal consumption and investment strategyπ=(π1,π2,…,πm).
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