| The investment portfolio is the hot issues in recent years. In the basic model, in-vestors generally will invest their own assets to risk products and risk-free products. In some early reference books, there are two common methods to study this problem:the max-imum principle and the dynamical programming principle. We will focus on the method of dynamical programming principle, which help us deduce Hamilton-Jacobi-Bellman func-tion(abbreviated as HJB function), and apply it in more complex investment environment.In this paper, first of all, on the basic investment consumption portfolio model, we con-sider inflation in the actual market. Basak and H.Yan[4], Campbell and T.Vuolteenaho[5], Cohen[6] have put forward the relationship between inflation and asset prices. The inflation itself also follows Brownian motion, which also corresponds to a stochastic differential equa-tion. On the other hand, the stock price is not always change continuously, so we introduce the Poisson process, generalizing the original model [8] further.As [8], we introduce the basic model firstly: X(t)=(?)f(t)Y0(t)+(?)(t)Y(t),(0.1) where (?)f(t) and (?)(t)is the weight of risk-free assets and risk assets, where (?)(t) and Y(t) are n dimensional vectors. The change of financing situation of assets is: dX(t)=(?)f(t)dY0(t)+(?){t)dY(t)-C(t)dt,(0.2) where C(t) is the instantaneous nominal consumption rate. With the market information completive, assets process can be written as dX(t)=r(t)X(t)dt+X(t)π(t)σ(t)[dw(t)+θ(t)dt]-C(t)dt,(0.3) note that π=(π1,...,πn)is the weight of risk assets, that is the proportion of investment in n stocks.In fact, the value of the asset is changed over time, and now assume that inflation follows a standard Brownian motion, the form is as follows:Firstly we define an objective function:In common, we give the state process:Its value function isIn order to obtain its HJB equation, we now put forward some assumptions on the systems (0.3)-(0.5):(Al)(U,d)is a Polish Space, T>0(A2)map b:[0,T]×Rn×U→Rn,σ:[0,T]×Rn×U→Rnxm,f:[0,T]×Rn×U→R,h:Rn→Ris uniformly continuous, exist a constant K>0so (?)(t,x,u)=b(t, x, u), σ(t, x, u),f(t, x, u), h(x),It is easy to have the following results by the main theorem in [11].Proposition0.1.(reference [11])Assume that (Al)(A2) hold,and value function V∈C1,2([O, T]: Rn) is given. Then V is a solution of HJB equation as follows: where From the above theorems,we can obtain the HJB equation as follows:Now we consider the new model:In the generalized model, we still give a theorem of HJB equation. Firstly, we define the state process: Its value function isNow assume that:(Al) b, σ, c is uniformly continuous about (t, x, u),and exists a constant K>0satisfy- ing:(A2)f, h is uniformly continuous about (t, x, u),and exist a constantK>0, and a con-tinuous increasing function (?)0:[0,∞)×[0,∞)→[0,∞),satisfying(?)o(τ,0)=0, τ(?)≥O,that makingIn this paper, we consider the inflation function, which makes the problem difficult to solve. The state process turn into (?)=(x, L), so we get the theorem:Theorem0.1. If(A1)’,(A2) are established, then the value function V∈C1,2([0, T]×Rn) is the solution of the function as follows:Here the generalized Hamilton function G is:By the above theorem, we get the main result for the value function:Theorem0.2. Considering the problems of optimal investment and consumption with infla-tion and jump diffusion, we get the exact HJB equation for the value function V. Finally, we give a specific utility function to discuss how investors determine the opti-mal investment and consumption strategy (C*, π*). For the problem of investment containing inflation and Poisson jump (0.7), we obtain the corresponding HJB equation (0.8) of objec-tive function (0.5). So that we can find the corresponding optimal solution.Theorem0.3. We consider the situation with the utility function being CRRA, that is, Ui=y1-φ/1-φ,i=1,2, where0<φ<1,,we have the following optimal investment and consump-tion strategy:C*=eL(eLVx)-1/φ; For the investment ratio π of the stock, when the market jump down to cheap price, we should invest less risk assets than jump-free situation; On the con-trary, if the stock market jump up to the high price, we should raise the proportion of risk assets. |
