Some Non-monotone Schemes for Time Dependent Hamilton–Jacobi–Bellman Equations in Stochastic Control

We introduce some approximation schemes for linear and fully non-linear diffusion equations of Bellman type. Based on modified high order interpolators, the schemes proposed are not monotone but one can prove their convergence to the viscosity solution of the problem. Some of these schemes are related to a scheme previously proposed without proof of convergence. Effective implementation of these schemes in a parallel framework is discussed. They are extensively tested on some simple test case, and on some difficult ones where theoretical results of convergence are not available.

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