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Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management

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  • Zuzana Macova
  • Daniel Sevcovic

Abstract

The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as well as quantitative point of view. We construct useful bounds of solution yielding estimates for the optimal value of the stock to bond proportion in the portfolio. Furthermore we construct asymptotic expansions of a solution in terms of a small model parameter. Finally, we perform sensitivity analysis of the optimal solution with respect to various model parameters and compare analytical results of this paper with the corresponding known results arising from time-discrete dynamic stochastic optimization model.

Suggested Citation

  • Zuzana Macova & Daniel Sevcovic, 2009. "Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management," Papers 0905.0155, arXiv.org, revised Nov 2009.
    • Handle: RePEc:arx:papers:0905.0155
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    References listed on IDEAS

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    1. Merton, Robert C. & Samuelson, Paul A., 1974. "Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods," Journal of Financial Economics, Elsevier, vol. 1(1), pages 67-94, May.
    2. Merton, Robert C., 1975. "Theory of Finance from the Perspective of Continuous Time," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 10(4), pages 659-674, November.
    3. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
    4. Young, H Peyton, 1990. "Progressive Taxation and Equal Sacrifice," American Economic Review, American Economic Association, vol. 80(1), pages 253-266, March.
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    Cited by:

    1. Naoyuki Ishimura & Daniel Sevcovic, 2011. "On traveling wave solutions to Hamilton-Jacobi-Bellman equation with inequality constraints," Papers 1108.1035, arXiv.org, revised May 2012.

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