  We propose a class of numerical schemes for nonlocal HJB variational
inequalities (HJBVIs) with monotone drivers. The solution and free boundary of
the HJBVI are constructed from a sequence of penalized equations, for which a
continuous dependence result is derived and 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 recursive consumption-portfolio allocation problem.
