  In this work, we propose a class of numerical schemes for solving semilinear
Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise
naturally from exit time problems of diffusion processes with controlled drift.
We exploit policy iteration to reduce the semilinear problem into a sequence of
linear Dirichlet problems, which are subsequently approximated by a multilayer
feedforward neural network ansatz. We establish that the numerical solutions
converge globally in the $H^2$-norm, and further demonstrate that this
convergence is superlinear, by interpreting the algorithm as an inexact Newton
iteration for the HJBI equation. Moreover, we construct the optimal feedback
controls from the numerical value functions and deduce convergence. The
numerical schemes and convergence results are then extended to HJBI boundary
value problems corresponding to controlled diffusion processes with oblique
boundary reflection. Numerical experiments on the stochastic Zermelo navigation
problem are presented to illustrate the theoretical results and to demonstrate
the effectiveness of the method.
