Abstract:

It is often difficult to solve dynamic pricing problems involving multiple state variables. Therefore, an improved collocation method is proposed by introducing multidimensional interpolation and tensor product, and the convergence of the improved collocation method is theoretically analyzed. The improved collocation method is applied to solve the dynamic economic problem with multiple state variables. Besides, the practical application of the improved collocation method is given and the improved method is applied to solve the dynamic pricing problem with finite memory. In addition, the sequential state transference equation is constructed by comparing the difference between the Chebyshev grids of state variables and its approximated values, and the optimal price path is simulated based on sequential state transference. The numerical analysis shows that the improved collocation method is faster in converging to steady state than the linear-quadratic approximation method.

Key words: dynamic pricing, finite memory, dynamic programming, collocation method, multidimensional interpolation, tensor product

摘要：

涉及多个状态变量的动态定价问题往往难以求解。基于此，引入多维插值与张量积，提出一种改进的collocation方法用于求解多状态变量的动态优化问题，并从理论上分析了改进后的collocation方法的收敛性。将改进的collocation方法应用于求解多个状态变量的动态经济问题。给出了改进后的collocation方法的实际运用，应用于解决有限记忆的动态定价问题。利用状态变量的切比雪夫格点与其估计值构造出时序的状态转移，并依据状态转移模拟最优价格路径。通过数值分析表明，改进后的collocation方法比线性二次逼近法更快的收敛于稳态。

关键词: 动态定价, 有限记忆, 动态规划, collocation方法, 多维插值, 张量积