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Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

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  • Sona Kilianova
  • Daniel Sevcovic
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    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.

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    File URL: http://arxiv.org/pdf/1307.3672
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1307.3672.

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    Date of creation: Jul 2013
    Date of revision: Jul 2013
    Handle: RePEc:arx:papers:1307.3672

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    1. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers, Columbia - Graduate School of Business 95-08, Columbia - Graduate School of Business.
    2. Paul Milgrom & Ilya Segal, 2002. "Envelope Theorems for Arbitrary Choice Sets," Econometrica, Econometric Society, Econometric Society, vol. 70(2), pages 583-601, March.
    3. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-57, August.
    4. Tourin, Agnes & Zariphopoulou, Thaleia, 1994. "Numerical Schemes for Investment Models with Singular Transactions," Computational Economics, Society for Computational Economics, Society for Computational Economics, vol. 7(4), pages 287-307.
    5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780195102680, October.
    6. R. C. Merton, 1970. "Optimum Consumption and Portfolio Rules in a Continuous-time Model," Working papers, Massachusetts Institute of Technology (MIT), Department of Economics 58, Massachusetts Institute of Technology (MIT), Department of Economics.
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