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Semismooth Newton methods with a shooting-like technique for solving a constrained free-boundary HJB equation. (English) Zbl 1459.91160
Summary: This paper describes novel numerical methods for solving a constrained free-boundary Hamilton-Jacobi-Bellman (HJB) equation. The equation comes from an optimal dividend problem within financial insurance and has a classical solution under strict assumptions. However the numerical approach is significant since the problem can be classified as a constrained variational inequality with free boundary, for which numerical methods are few. We combine the semismooth Newton method with a shooting-like method to treat the non-smoothness and free boundary of the problem, respectively. We introduce two algorithms, differing only in their treatment of the inequality constraints; the first applies a sweeping method, the second utilizes projected Newton. For the latter, the associated convergence analysis is discussed. In the end, we show that the local convergence rate for this method is at least superlinear. The contribution of this paper is a numerical approach that can be applied to more general free boundary problems with first or second derivative constraints for which analytical solutions are not known.
MSC:
91G05 Actuarial mathematics
93E20 Optimal stochastic control
49M15 Newton-type methods
Software:
KELLEY
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