Approximation Schemes for Mixed Optimal Stopping and Control Problems with Nonlinear Expectations and Jumps

TY - JOUR

T1 - Approximation Schemes for Mixed Optimal Stopping and Control Problems with Nonlinear Expectations and Jumps

AU - Dumitrescu, Roxana

AU - Reisinger, Christoph

AU - Zhang, Yufei

PY - 2019/7/5

Y1 - 2019/7/5

N2 - 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.

AB - 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.

KW - Approximation schemes

KW - Jump processes

KW - Nonlinear expectations

KW - Optimal stopping

KW - Piecewise constant policy timestepping

KW - Stochastic control

UR - http://www.scopus.com/inward/record.url?scp=85068862636&partnerID=8YFLogxK

U2 - 10.1007/s00245-019-09591-0

DO - 10.1007/s00245-019-09591-0

M3 - Article

AN - SCOPUS:85068862636

SN - 0095-4616

VL - 2019

JO - APPLIED MATHEMATICS AND OPTIMIZATION

JF - APPLIED MATHEMATICS AND OPTIMIZATION

ER -