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Optimal investment and consumption models with non-linear stock dynamics

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  • Thaleia Zariphopoulou

Abstract

We study a generalization of the Merton's original problem of optimal consumption and portfolio choice for a single investor in an intertemporal economy. The agent trades between a bond and a stock account and he may consume out of his bond holdings. The price of the bond is deterministic as opposed to the stock price which is modelled as a diffusion process. The main assumption is that the coefficients of the stock price diffusion are arbitrary nonlinear functions of the underlying process. The investor's goal is to maximize his expected utility from terminal wealth and/or his expected utility of intermediate consumption. The individual preferences are of Constant Relative Risk Aversion (CRRA) type for both the consumption stream and the terminal wealth. Employing a novel transformation, we are able to produce closed form solutions for the value function and the optimal policies. In the absence of intermediate consumption, the value function can be expressed in terms of a power of the solution of a homogeneous linear parabolic equation. When intermediate consumption is allowed, the value function is expressed via the solution of a non-homogeneous linear parabolic equation. Copyright Springer-Verlag Berlin Heidelberg 1999

Suggested Citation

  • Thaleia Zariphopoulou, 1999. "Optimal investment and consumption models with non-linear stock dynamics," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 50(2), pages 271-296, October.
    • Handle: RePEc:spr:mathme:v:50:y:1999:i:2:p:271-296
    DOI: 10.1007/s001860050098
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    Cited by:

    1. L. A. Bordag & I. P. Yamshchikov & D. Zhelezov, 2015. "Portfolio optimization in the case of an asset with a given liquidation time distribution," Post-Print hal-01186961, HAL.
    2. Li, Zhongfei & Zeng, Yan & Lai, Yongzeng, 2012. "Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 191-203.
    3. Benjamín Vallejo Jiménez & Francisco Venegas Martínez, 2017. "Optimal consumption and portfolio rules when the asset price is driven by a time-inhomogeneous Markov modulated fractional Brownian motion with," Economics Bulletin, AccessEcon, vol. 37(1), pages 314-326.
    4. Jean-Pierre Fouque & Ruimeng Hu, 2018. "Portfolio Optimization under Fast Mean-reverting and Rough Fractional Stochastic Environment," Papers 1804.03002, arXiv.org, revised Jan 2019.
    5. K. Charalambous & S. Kontogiorgis & C. Sophocleous, 2021. "The Lie symmetry approach on (1+2)-dimensional financial models," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-17, August.
    6. Robert Cox Merton & Francisco Venegas-Martínez, 2021. "Tendencias y perspectivas de la ciencia financiera: Un artículo de revisión," Remef - Revista Mexicana de Economía y Finanzas Nueva Época REMEF (The Mexican Journal of Economics and Finance), Instituto Mexicano de Ejecutivos de Finanzas, IMEF, vol. 16(1), pages 1-15, Enero - M.
    7. Shen, Yang & Zeng, Yan, 2015. "Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 118-137.
    8. Ruimeng Hu, 2018. "Asymptotic Optimal Portfolio in Fast Mean-reverting Stochastic Environments," Papers 1803.07720, arXiv.org, revised Jan 2019.
    9. Robert Cox Merton & Francisco Venegas-Martínez, 2021. "Financial Science Trends and Perspectives: A Review Article," Remef - Revista Mexicana de Economía y Finanzas Nueva Época REMEF (The Mexican Journal of Economics and Finance), Instituto Mexicano de Ejecutivos de Finanzas, IMEF, vol. 16(1), pages 1-15, Enero - M.
    10. Maxim Bichuch & Jean-Pierre Fouque, 2019. "Optimal Investment with Correlated Stochastic Volatility Factors," Papers 1908.07626, arXiv.org, revised Nov 2022.
    11. Jean-Pierre Fouque & Ruimeng Hu, 2017. "Optimal Portfolio under Fast Mean-reverting Fractional Stochastic Environment," Papers 1706.03139, arXiv.org, revised Feb 2018.
    12. Jean-Pierre Fouque & Ruimeng Hu, 2019. "Multiscale Asymptotic Analysis for Portfolio Optimization under Stochastic Environment," Papers 1902.06883, arXiv.org, revised Sep 2019.
    13. Oleksii Mostovyi & Mihai Sîrbu, 2019. "Sensitivity analysis of the utility maximisation problem with respect to model perturbations," Finance and Stochastics, Springer, vol. 23(3), pages 595-640, July.
    14. Holger Kraft & Mogens Steffensen, 2006. "Portfolio problems stopping at first hitting time with application to default risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 123-150, February.
    15. Guiyuan Ma & Song-Ping Zhu, 2022. "Revisiting the Merton Problem: from HARA to CARA Utility," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 651-686, February.
    16. Ljudmila A. Bordag & Ivan P. Yamshchikov & Dmitry Zhelezov, 2014. "Portfolio optimization in the case of an asset with a given liquidation time distribution," Papers 1407.3154, arXiv.org.
    17. Jean-Pierre Fouque & Ruimeng Hu, 2016. "Asymptotic Optimal Strategy for Portfolio Optimization in a Slowly Varying Stochastic Environment," Papers 1603.03538, arXiv.org, revised Nov 2016.
    18. Marco Piccirilli & Tiziano Vargiolu, 2018. "Optimal Portfolio in Intraday Electricity Markets Modelled by L\'evy-Ornstein-Uhlenbeck Processes," Papers 1807.01979, arXiv.org.
    19. Bakari, Sayef, 2022. "The Nexus between Domestic Investment and Economic Growth in Developed Countries: Do Exports matter?," MPRA Paper 114394, University Library of Munich, Germany.
    20. Ljudmila A. Bordag & Ivan P. Yamshchikov, 2015. "Optimization problem for a portfolio with an illiquid asset: Lie group analysis," Papers 1512.06295, arXiv.org.
    21. Henderson, Vicky, 2005. "Explicit solutions to an optimal portfolio choice problem with stochastic income," Journal of Economic Dynamics and Control, Elsevier, vol. 29(7), pages 1237-1266, July.
    22. Jean-Pierre Fouque & Ruimeng Hu, 2017. "Optimal Portfolio under Fractional Stochastic Environment," Papers 1703.06969, arXiv.org, revised Dec 2017.

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