Discrete time dynamic programming to solve dynamic portfolio choice models has three immanent issues: firstly, the curse of dimensionality prohibits more than a handful of continuous states. Secondly, in higher dimensions, even regular sparse grid discretizations need too many grid points for sufficiently accurate approximations of the value function. Thirdly, the models usually require continuous control variables, and hence gradient-based optimization with smooth approximations of the value function is necessary to obtain accurate solutions to the optimization problem. For the first time, we enable accurate and fast numerical solutions with gradient-based optimization while still allowing for spatial adaptivity using hierarchical B-splines on sparse grids. When compared to the standard linear bases on sparse grids or finite difference approximations of the gradient, our approach saves an order of magnitude in total computational complexity for a representative dynamic portfolio choice model with varying state space dimensionality, stochastic sample space, and choice variables.

解决动态投资组合选择模型的离散时间动态规划具有三个内在的问题：首先，维数的诅咒禁止多个连续状态。其次，在更高的维度上，即使规则的稀疏网格离散化也需要太多的网格点，才能足够精确地近似值函数。第三，模型通常需要连续的控制变量，因此必须使用基于梯度的优化和值函数的平滑近似才能获得优化问题的精确解。我们首次通过基于梯度的优化实现了精确而快速的数值解，同时仍然允许在稀疏网格上使用分层B样条实现空间适应性。