  We propose a class of numerical schemes for mixed optimal stopping and
control of processes with infinite activity jumps and where the objective is
evaluated by a nonlinear expectation. Exploiting an approximation by switching
systems, piecewise constant policy timestepping reduces the problem to nonlocal
semi-linear equations with different control parameters, uncoupled over
individual time steps, which we solve by fully implicit monotone approximations
to the controlled diffusion and the nonlocal term, and specifically the
Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a
comparison principle for the switching system and demonstrate the convergence
of the schemes, which subsequently gives a constructive proof for the existence
of a solution to the switching system. Numerical experiments are presented for
a recursive utility maximization problem to demonstrate the effectiveness of
the new schemes.
