A Hamilton-Jacobi-Bellman approach to optimal trade execution
Applied Numerical Mathematics
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
SIAM Journal on Numerical Analysis
Penalty Methods for the Solution of Discrete HJB Equations—Continuous Control and Obstacle Problems
SIAM Journal on Numerical Analysis
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In order to ensure convergence to the viscosity solution, the standard method for discretizing Hamilton-Jacobi-Bellman partial differential equations uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic equations, we have to maximize a possibly discontinuous objective function at each node. Nevertheless, convergence of the overall iteration can be guaranteed. Numerical experiments on two examples from the finance literature show higher rates of convergence for this approach compared to the use of forward/backward differencing only.