Knight Uncertain Environment, The Distribution Of Difference To Measure The Number Of Economics, And Sound Decision-making

Under Knightian uncertainty, it is difficult for decision-makers to predict future economy states with a single probability distribution, but with a set of subjective probability measures (that is, a priori probability measures set). There are two core issues of robust control and decision-making, one is how to describe the priori probability measures set of decision-maker, the other is how to make decision based on multiple priori probability measures. This paper make theoretical study closely around the above issues.The first part researched probability distribution divergence measures to describe the multiple priori probability measures set. Firstly, three basis information measures, as entropy, relative entropy and Fisher information, were introduced and the relation among them was discussed to make preparation for the later study.Secondly, the paper studied the relative entropy specifically which was an indicator used commonly, where, analyzed the relative entropy of Normal distributions family and its convergence, established a space structure of Normal distributions and a homeomorphic map from probability measure space to its parameter space, obtained approximation of relative entropy by the Taylor formula, calculated the relative entropy of common static finacial distributions including the normal distribution, the lognormal distribution, t-distribution and Pareto distribution, and calculated the relative entropy of some special stochastic process, such as the logarithm of stock prices based on geometry Brownian motion, and stock prices when drift parameter was distorted in the diffusion process. These studies will promote the application in robust decision-making under Knightian uncertainty.Thirdly, because it is often difficult to get a specific probability distribution, and relatively easy to obtain the moments of distributions, a divergence measure between distributions was established depending only on moments of the distributions and not on the specific forms of distributions, and offers a powerful tool to measure divergence between distributions.The second part of the paper studied the robust decision-making methods based on multiple priori probability measures under Knightian uncertainty. First, it summarized the Multiple-Priors Utility, two classical models of the max-min expected utility and maximizing multiplier utility models. Second, considering the decision makers under Knightian uncertainty always get incomplete information, we introduced the maximum entropy principal and minimum relative entropy principal. Then, in order to avoid big deviation between the reference distribution and the unknown true distribution, we proposed a robust decision-making model based on the maximum entropy and the max-min expected utility, that is, choosing the maximum entropy distribution as the reference distribution and then maximizing the minimum expected utility under the entropy-constrained ball (or maximizing multiplier utility).Knightian uncertainty is a hot spot, and every investor may face in decision-making. These theoretical reseach will promote the robust control and decision-making under Knightian uncertainty.