AI 中文总结
研究具有期望约束的效用最大化问题，在一致椭圆和退化情形下分别提出方法，通过新截断论证及添加噪声项得到逼近序列并证明收敛，首次全面分析此类问题，用神经网络进行数值求解与误差估计。
AI 中文摘要
如Bouchard等人（2010年）和Bouchard与Nutz（2014年）所述，我们研究具有期望约束的效用最大化问题。首先考虑一致椭圆情形，其中与期望约束相关的内生状态边界被证明是光滑的，这能为最优控制问题的值函数在该边界上导出适当的狄利克雷条件。接着在期望约束的鞅表示中提出新的截断论证，得到一个具有比较性质的辅助偏微分方程组的逼近序列，并证明其收敛到初始最优控制问题。在退化情形下，通过添加小噪声项恢复一致椭圆性提出另一种逼近并证明收敛性。据我们所知，首次对这类控制问题进行全面分析，为数值方案的使用打开大门。在一个简单例子中用神经网络进行数值求解，并通过神经网络方法估计数值误差。
英文摘要
As in Bouchard et al. (2010) and Bouchard and Nutz (2014), we study a utility maximization problem with expectation constraint. We first consider a uniformly elliptic case in which the endogenous state boundary associated with the constraint in expectation is proved to be smooth. This allows one to derive a proper Dirichlet condition for the value function of the optimal control problem on this boundary. We then propose a new truncation argument in the martingale representation of the expectation constraint. This leads to an approximating sequence of auxiliary systems of PDEs for which comparison holds. Convergence to the initial optimal control problem is proved. In the degenerate case, we propose another approximation which consists in adding a small noise term to recover uniformly ellipticity. Convergence is also proved. To the best of our knowledge, it is the first time that a full analysis is performed for such control problems, so as to open the doors to the use of numerical schemes. Numerical resolution in a toy example is performed using neural networks. It is complemented by an estimation of the numerical error, also performed by using a neural network approach.
