We propose a class of numerical schemes for nonlocal HJB variational inequalities (HJBVIs) with monotone nonlinearities arising from mixed optimal stopping and control of processes with infinite activity jumps, where the objective is specified by a monotone recursive preference. The solution and free boundary of the HJBVI are constructed from a sequence of penalized equations, for which the penalization error is estimated. The penalized equation is then discretized by a class of semi-implicit monotone approximations. We present a novel analysis technique for the well-posedness of the discrete equation, and demonstrate the convergence of the scheme, which subsequently gives a constructive proof for the existence of a solution to the penalized equation and variational inequality. We further propose an efficient iterative algorithm with local superlinear convergence for solving the discrete equation. Numerical experiments are presented for an optimal investment problem under ambiguity and a two-dimensional recursive consumption-portfolio allocation problem.

我们为非局部HJB变分不等式（HJBVI）提出了一类数值方案，该方案由于混合最优停止和具有无限活动跳跃的过程的控制而产生单调非线性，其目标由单调递归偏好指定。HJBVI的解和自由边界是由一系列罚分方程构成的，为此估计了罚分误差。然后，通过一类半隐式单调逼近来离散该惩罚方程。我们为离散方程的适定性提出了一种新颖的分析技术，并证明了该方案的收敛性，随后为有损方程和变分不等式的解的存在提供了建设性的证明。我们还提出了一种具有局部超线性收敛的有效迭代算法来求解离散方程。针对歧义下的最优投资问题和二维递归消费-投资组合分配问题，进行了数值实验。