A transformation method for solving Hamilton-Jacobi-Bellman equation for constrained dynamic stochastic optimal allocation problem

Authors

  • Daniel Sevcovic Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava
  • Sona Kilianova Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava

DOI:

https://doi.org/10.21914/anziamj.v55i0.6816

Keywords:

Hamilton--Jacobi--Bellman equation, Riccati transformation, quasi-linear parabolic equation, finite volume approximation scheme, traveling wave solution

Abstract

In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore 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 traveling 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 application of the method. doi:10.1017/S144618111300031X

Author Biographies

Daniel Sevcovic, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava

Department of Applied Mathematics and Statistics

Sona Kilianova, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava

Department of Applied Mathematics and Statistics

Published

2014-04-03

Issue

Section

Articles for Printed Issues