基于非线性期望理论的稳健货币政策与财政政策设计

The economy of China is currently affected by the international financial crisis and domestic inflation volatility. It also faces the pressure of deflation and the possibility of falling into the "liquidity trap". We plan to investigate the optimal design of monetary and fiscal policies by stochastic control model in continuous time to avoid the above mentioned risks. In order to deal with the uncertainty of the model, when we design the optimal monetary and fiscal policies, we hope that the obtained policies are robust. The nonlinear expectation theory (the g-expectation and G- expectation) proposed by Peng is employed to construct the continuous models which describe the uncertainty and to analyze the optimal robust monetary and fiscal policies which are based on the inflation and output models. We found that stochastic models in continuous time are described by a new kind of forward backward stochastic differential equations (FBSDE) on infinite horizon. The uniqueness and existence of the solutions of these equations is unknown by the existed theories. So it is an interesting mathematical problem by itself. The properties of the optimal monetary and fiscal policies are deduced by the stochastic maximum principle. We hope that this project will be helpful for the decision makers when they design the robust monetary and fiscal policies.

我国目前受到国际金融危机、国内通胀波动的影响,面临着通货紧缩的压力,同时存在着陷入“流动性陷阱”的可能性。本项目探讨如何运用连续时间随机控制模型来分析并得到最优的货币政策和财政政策，以防范上述风险。为了应对模型的不确定性，在进行最优的货币财政政策设计时，需要得到的政策同时具有稳健性。本项目以彭实戈教授提出的非线性期望理论（g-期望和G-期望）为数学框架，构建具有刻画不确定性的连续时间模型，得到并分析基于通胀和产出模型的最优的稳健货币财政政策。而且，我们发现连续时间随机情形下的模型需要用一类新型的无穷区间上的正倒向随机微分方程(FBSDE)来刻画。这类方程无法用现有的理论判断其解的存在唯一性，因而其本身也是一个有趣的数学问题。我们以随机最大值原理为工具，推导出最优货币财政政策所满足的性质。希望本项目的研究，会对宏观经济政策的决策者在设计稳健的货币财政政策时有所帮助。

本项目围绕如何运用连续时间随机控制理论来分析并得到最优的投资策略和货币财政政策，以防范金融风险。为了应对模型的不确定性，在进行最优策略设计时，需要得到的政策同时具有稳健性。本项目以正倒向随机微分方程和彭实戈教授提出的非线性期望理论为理论工具，得到（无穷区间上的）完全耦合FBSDE解的存在唯一性定理，从而保证连续时间最优投资组合、货币财政政策模型的适定性；推导出基于完全耦合FBSDE的递归效用优化问题的随机最大值原理。对G-布朗运动驱动的模型，得到保证解的存在唯一的条件和相应的稳健的随机最大值原理,并应用于递归效用下的稳健最优投资组合和最优货币财政政策的研究。本项目的研究会对金融市场中的投资者构建最优投资组合和宏观经济政策的决策者设计稳健的货币财政政策提供依据。