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Numerical Solution of the Hamilton-Jacobi-Bellman Equation for Technical and Financial Applications of Stochastic Optimal Control


H. Peyrl

TU Graz, Diploma Thesis at the Measurement and Control Laboratory, ETH Zürich; Institute of Automatic Control.

The range of stochastic optimal feedback control problems covers a variety of physical, biological, economic and management systems. Since the arising Hamilton-Jacobi-Bellman (HJB) equation can be solved analytically only for simple examples, this diploma thesis provides a numerical solution of the HJB equation for stochastic systems with non-linear state equations. The computation's difficulty has its root in the nature of the HJB equation being a second-order partial differential equation (PDE) that is in addition coupled with an optimization. The problem of solving both PDE and optimization simultanously is overcome by applying a successive approximation algorithm. The algorithm is an iterative approach that separates the optimization from the boundary value problem and is iterated until convergence. To compute the numerical solution of the arising PDEs, solvers based on finite difference schemes, capable to handle problems up to three dimensions are developed. The corresponding optimization problems are solved for selected case studies with respect to possible control law constraints. After some comparisons with analytical solutions to validate the quality of the successive approximation algorithm, its usefulness is shown by computing numerical solutions of two financial portfolio examples that cannot be solved analytically. Finally, statistic evaluations back up the power of the used methods.

Further Information

Type of Publication:

(12)Diploma/Master Thesis

F. Herzog

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% Autogenerated BibTeX entry
@PhdThesis { Xxx:2003:IFA_644
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