随机最优控制中的半经典逼近I。投资组合构建问题
Semiclassical approximation in stochastic optimal control I. Portfolio construction problem
摘要：
这是第一次在一系列的文件中，我们研究了一个有效的近似方案，解决哈密顿-雅可比-贝尔曼方程的多维问题，在随机控制理论。该方法是一个组合的WKB风格的渐近展开的价值函数，它减少了二阶HJB偏微分方程的一阶偏微分方程的层次，其次是一个数值算法来解决的前几个产生的一阶偏微分方程。该方法适用于自由度相对较大的随机系统，并且似乎不会受到维数灾难的影响。使用适度计算资源的方法的计算机代码实现基本上实时运行。我们应用该方法来解决一般的投资组合建设问题。
英文摘要：
This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order HJB partial differential equation to a hierarchy of first order PDEs, followed by a numerical algorithm to solve the first few of the resulting first order PDEs. This method is applicable to stochastic systems with a relatively large number of degrees of freedom, and does not seem to suffer from the curse of dimensionality. Computer code implementation of the method using modest computational resources runs essentially in real time. We apply the method to solve a general portfolio construction problem.
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