On Study Of Optimal Consumption And Portfolio With Inflation And Stochastic Income Under Knight Uncertainty

As one of the most fundamental problems in financial mathematics, the study of optimal consumption and portfolio has always been concerned by scholars at home and abroad.This paper mainly studies the problem of the effect of inflation and stochas-tic income on optimal consumption and portfolio under knight uncertainty, when risky asset prices follow a geometric Brownian motion in financial market. Firstly, we describe an inflation process and a wealth process in a financial market when the inflation volatility is a constant, and we describe the investor’s preference with stochastic differential utility. By using the principle of dynamic programming, we study optimal consumption and portfolio under inflation when investor has unit elasticity of intertemporal substitution of consumption and derive an exact solution according to the assumed value function, then to analyze the effect of the inflation volatility on optimal portfolio.Secondly, as the investor’s elasticity of intertemporal substitution of consump-tion may differ from unit, we get an exact solution when we take it as unit; when elasticity of intertemporal substitution is different from unit, we get a nonlinear or-dinary differential equation, so we have to use approximate substitution, then we get an approximate solution of optimal consumption and portfolio and analyze the effect of reversion parameter on portfolio choice. By contrast, we conclude that the same optimal portfolio and different consumption strategy are obtained when investor’s elasticity of intertemporal substitution varies.Thirdly, the inflation volatility in financial market may be not fixed, but take value in an interval, we take the effect of the uncertainty of inflation volatility on portfolio into account. As investor make decision based on the minimal utility, the demand for the risky asset can be decomposed into three components:myopic and inflation hedging demands and ambiguity demand. The effect of the parameters on investor’s decision is analyzed.Finally, as stochastic income can raise investor’s wealth level, optimal portfolio choice can be decomposed into two components:myopic and income hedging de-mands. When income volatility varies, we conclude that:optimal consumption and portfolio choice can be made up of myopic and income hedging demands and ambi-guity demand; the higher wealth, the higher consumption; income hedging demand always be negative when the correlation of stochastic income uncertainty and the asset’s return is positive.In the paper, Some mathematical methods are mainly utilized to study op-timal consumption and portfolio, such as optimal stochastic control and dynamic programming approach, etc.