Approximation schemes for mixed optimal stopping and control problems with nonlinear expectations and jumps
For researchers in stochastic control and mathematical finance, this provides a convergent numerical method for a class of problems that previously lacked such schemes.
The paper proposes numerical schemes for mixed optimal stopping and control problems with jumps and nonlinear expectations, proving convergence and demonstrating effectiveness on a recursive utility maximization problem.
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.