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We propose a class of numerical schemes for nonlocal HJB variationalinequalities (HJBVIs) with monotone drivers. The solution and free boundary ofthe HJBVI are constructed from a sequence of penalized equations, for which acontinuous dependence result is derived and the penalization error isestimated. The penalized equation is then discretized by a class ofsemi-implicit monotone approximations. We present a novel analysis techniquefor the well-posedness of the discrete equation, and demonstrate theconvergence of the scheme, which subsequently gives a constructive proof forthe existence of a solution to the penalized equation and variationalinequality. We further propose an efficient iterative algorithm with localsuperlinear convergence for solving the discrete equation. Numericalexperiments are presented for an optimal investment problem under ambiguity anda recursive consumption-portfolio allocation problem.

A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone drivers