Numerical methods for stochastic control problems in continuous time
SIAM Journal on Control and Optimization
Numerical methods for stochastic control problems in continuous time
Numerical methods for stochastic control problems in continuous time
Quadratic Convergence for Valuing American Options Using a Penalty Method
SIAM Journal on Scientific Computing
Convergence Properties of Policy Iteration
SIAM Journal on Control and Optimization
Tools for Computational Finance (Universitext)
Tools for Computational Finance (Universitext)
Automatica (Journal of IFAC)
Maximal Use of Central Differencing for Hamilton-Jacobi-Bellman PDEs 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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We present a simple and easy-to-implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, classical finite difference discretizations can be shown to converge to the unique viscosity solutions of the considered problems. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of $O(1/\rho)$, where $\rho0$ is the penalty parameter, and we show that an iterative scheme can be used to solve the penalized discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.