论文标题
一种基于SMP的算法，用于通过深度学习解决受限的实用性最大化问题
An SMP-Based Algorithm for Solving the Constrained Utility Maximization Problem via Deep Learning
论文作者
论文摘要
我们考虑了凸约限制下关于理论结果的实用性最大化问题，这些结果允许使用深度学习技术的算法求解器的制定。特别是对于随机系数的情况，我们证明了一个随机最大原理（SMP），该功能还具有$ \ mathrm {id} _ {\ Mathbb {\ mathbb {r}^{+}^{+}} \ cdot u'''不一定是不一定的，在此期间不一定要努力（Z），该公司不一定要努力（Z）。我们将此SMP与强双重属性一起使用来定义一种新算法，我们称之为深度Primal SMP算法。数值示例说明了所提出的算法的有效性 - 特别是对于高维问题和随机系数的问题，这些问题依赖或满足其自身的SDE。此外，我们针对受约束问题的数值实验表明，新型的深度原始SMP算法克服了深层SMP算法（参见Davey和Zheng（2021））的弱点，错误地产生了相应的无约束问题的值。此外，与Davey和Zheng（2021）的深层控制的2BSDE算法相反，该算法也适用于依赖路径系数的问题。当深层SMP算法甚至在我们的许多研究问题中产生最准确的结果，我们可以强烈建议其使用情况。此外，我们提出了一个基于时期的学习程序，从而进一步改善了算法的结果。实现半循环网络体系结构为控制过程是一个有价值的进步。
We consider the utility maximization problem under convex constraints with regard to theoretical results which allow the formulation of algorithmic solvers which make use of deep learning techniques. In particular for the case of random coefficients, we prove a stochastic maximum principle (SMP), which also holds for utility functions $U$ with $\mathrm{id}_{\mathbb{R}^{+}} \cdot U'$ being not necessarily nonincreasing, like the power utility functions, thereby generalizing the SMP proved by Li and Zheng (2018). We use this SMP together with the strong duality property for defining a new algorithm, which we call deep primal SMP algorithm. Numerical examples illustrate the effectiveness of the proposed algorithm - in particular for higher-dimensional problems and problems with random coefficients, which are either path dependent or satisfy their own SDEs. Moreover, our numerical experiments for constrained problems show that the novel deep primal SMP algorithm overcomes the deep SMP algorithm's (see Davey and Zheng (2021)) weakness of erroneously producing the value of the corresponding unconstrained problem. Furthermore, in contrast to the deep controlled 2BSDE algorithm from Davey and Zheng (2021), this algorithm is also applicable to problems with path dependent coefficients. As the deep primal SMP algorithm even yields the most accurate results in many of our studied problems, we can highly recommend its usage. Moreover, we propose a learning procedure based on epochs which improved the results of our algorithm even further. Implementing a semi-recurrent network architecture for the control process turned out to be also a valuable advancement.
