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A TRANSFORMATION METHOD FOR SOLVING THE HAMILTON–JACOBI–BELLMAN EQUATION FOR A CONSTRAINED DYNAMIC STOCHASTIC OPTIMAL ALLOCATION PROBLEM

Part of: Hamiltonian and Lagrangian mechanics Parabolic equations and systems Mathematical economics Asymptotic theory

Published online by Cambridge University Press:  10 October 2013

S. KILIANOVÁ  and
D. ŠEVČOVIČ
S. KILIANOVÁ*
Affiliation:
Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia email [email protected]
D. ŠEVČOVIČ
Affiliation:
Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia email [email protected]
*
✉[email protected]
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Abstract

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We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

Keywords

Hamilton–Jacobi–Bellman equationRiccati transformationquasilinear parabolic equationfinite volume approximation schemetravelling wave solution

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 34E05: Asymptotic expansions 70H20: Hamilton-Jacobi equations 91B16: Utility theory 91B70: Stochastic models 90C15: Stochastic programming
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 1 , July 2013 , pp. 14 - 38
Copyright
Copyright ©2013 Australian Mathematical Society 

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