Optimal Control with Expectation Constraint in a Smooth Boundary Case

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.