ISSN 0253-2778

CN 34-1054/N

open
Free to Publish. Free to Read.
Open AccessOpen Access JUSTC Original Paper

Optimization of dynamic portfolio under model uncertainty

  • Department of Financial Engineering, Anhui Polytechnic University, Wuhu 241000, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2014.03.005
  • Received Date: 16 July 2012
  • Accepted Date: 23 October 2012
  • Rev Recd Date: 23 October 2012
  • Publish Date: 30 March 2014
    Abstract

  • Abstract

    The problem of optimal portfolio under model uncertainty and a general semimartingale market was studied. First, a solution to the investment problem was obtained using the martingale method and the dual theory. It was proven that under appropriate assumptions a unique solution to the investment problem exists and is characterized. Then, the value functions of the primal and dual problem are convex conjugate functions. Finally, a diffusion-jump-model was considered where the coefficients depend on the state of a Markov chain and the investor is uncertain about the intensity of the underlying Poisson process. For an agent with logarithmic utility function, the stochastic control method was adopted to derive the Hamilton-Jacobi-Bellmann-equation. Furthermore, the solution of the dual problem can be determined and it was shown how the optimal portfolio can be explicitly computed.

    Abstract

    The problem of optimal portfolio under model uncertainty and a general semimartingale market was studied. First, a solution to the investment problem was obtained using the martingale method and the dual theory. It was proven that under appropriate assumptions a unique solution to the investment problem exists and is characterized. Then, the value functions of the primal and dual problem are convex conjugate functions. Finally, a diffusion-jump-model was considered where the coefficients depend on the state of a Markov chain and the investor is uncertain about the intensity of the underlying Poisson process. For an agent with logarithmic utility function, the stochastic control method was adopted to derive the Hamilton-Jacobi-Bellmann-equation. Furthermore, the solution of the dual problem can be determined and it was shown how the optimal portfolio can be explicitly computed.
    • FullText(HTML)
  • loading
    • References(28)
  • [1]
    Merton R C. Lifetime portfolio selection under uncertainty: The continuous time case[J]. Review of Economics and Statistics, 1969, 51: 247-257.
    [2]
    Merton R C. Optimum consumption and portfolio rules in a continuous time model[J]. Journal of Economic Theory, 1971, 3: 373-413.
    [3]
    Knight F H. Risk, Uncertainty and Profit[M]. Boston: Houghton Mifflin, 1921.
    [4]
    Gilboa I, Schmeidler D. Maxim expected utility with non-unique prior[J]. Journal of Mathematical Economics, 1989, 18:141-153.
    [5]
    Kramkov D, Schachermayer W. The asymptotic elasticity of utility functions and optimal investment in incomplete markets[J]. Annals of Applied Probability, 1999, 9(3): 904-950.
    [6]
    Cvitanic J, Schachermayer W, Wang H. Utility maximization in incomplete markets with random endowment[J]. Finance and Stochastics, 2001, 5(2): 259-272.
    [7]
    Karatzas I, Zitkovic G. Optimal consumption from investment and random endowment in incomplete semimartingale markets[J]. Annals of Probability, 2003, 31(4): 1 821-1 858.
    [8]
    Hugonnier J, Kramkov D. Optimal investment with random endowments in incomplete markets[J]. Annals of Applied Probability, 2004, 14(2): 845-864.
    [9]
    Schied A. Optimal investments for risk- and ambiguity-averse preferences: A duality approach[J]. Finance and Stochastics, 2007, 11(1): 107-129.
    [10]
    Schied A, Wu C T. Duality theory for optimal investments under model uncertainty[J]. Statistics Decisions, 2005, 23(3): 199-217.
    [11]
    Gundel A. Robust utility maximization for complete and incomplete market models[J]. Finance and Stochastics, 2005, 9(2): 151-176.
    [12]
    Duffie D, Zariphopoulou T. Optimal investment with undiversifiable income risk[J]. Mathematical Finance, 1993, 3:135-148.
    [13]
    Quenez M C. Optimal portfolio in a multiple-priors model[J]. Seminar on Stochastic Analysis, Random Fields and Applications Ⅳ, Progress in Probability, 2004, 58: 291-321.
    [14]
    Becherer D. Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging[J]. Annals of Applied Probability, 2006, 16(4): 2 027-2 054.
    [15]
    Muller M. Market completion and robust utility maximization[D]. Berlin: Humboldt University, 2005.
    [16]
    Bauerle N, Rieder U. Portfolio optimization with jumps and unobservable intensity process[J]. Mathematical Finance, 2007, 2(17): 205-224.
    [17]
    Hernandez-Hernandez D, Schied A. A control approach to robust utility maximization with logarithmic utility and time consistent penalties[J]. Stochastic Processes and Applications, 2007, 117(8): 980-1 000.
    [18]
    Fei W Y. Optimal portfolio choice based on α-MEU under ambiguity [J]. Stochastic Models, 2009, 25: 455-482.
    [19]
    Fei W Y. Optimal consumption and portfolio choice with ambiguity and anticipation [J]. Information Sciences, 2007, 177: 5 178-5 190.
    [20]
    Fei Weiyin, Li Shujuan. Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J]. Chinese Journal of Engineering Mathematics, 2012, 29(6): 799-806.
    费为银, 李淑娟. Knight 不确定下带通胀的最优消费和投资模型研究[J]. 工程数学学报, 2012, 29(6): 799-806.
    [21]
    Yang Zhaojun. Maximizing the expected utility from terminal wealth under the case of partial information[J]. Control Theory & Applications, 2006, 20(2): 11-13.
    杨招军. 部分信息下极大化终止时刻期望效用[J]. 控制理论与应用, 2006, 20(2): 11-13.
    [22]
    Xia Dengfeng, Fei Weiyin, Liu Hongjian. On study of optimal investment with ambiguity and anticipation under fluctuated discounting rate[J]. Chinese Journal of Applied Probability and Statistics, 2010, 26(3): 270-276.
    夏登峰, 费为银, 刘宏建. 变折现率下带含糊和预期的投资问题研究[J]. 应用概率统计, 2010, 26(3): 270-276.
    [23]
    Han Liyan, Pan Min. Knightian uncertainty based option pricing with stochastic volatility[J]. Systems Engineering: Theory & Practice, 2012, 32(6): 1 175-1 183.
    韩立岩, 泮敏. 基于奈特不确定性随机波动率期权定价[J]. 系统工程理论与实践, 2012, 32(6): 1 175-1 183.
    [24]
    Li Juan, Fei Weiyin, Shi Xueqin, et al. Optimal trading strategy under disordered asset return and Knightian uncertainty[J]. Applied Mathematics A Journal of Chinese Universities, 2013, 28(1): 13-22.
    李娟, 费为银, 石学芹,等. 奈特不确定下资产收益率发生紊乱的最优投资策略[J]. 高校应用数学学报, 2013, 28(1): 13-22.
    [25]
    Fei Weiyin, Chen Chao, Liang Yong. Optimal consumption-portfolio and retirement problem with disutility under Knightian Uncertainty[J]. Chinese Journal of applied probability and statistics, 2013, 29(1): 53-63.
    费为银, 陈超, 梁勇. Knight 不确定下考虑负效用的消费和投资问题研究[J]. 应用概率统计, 2013, 29(1): 53-63.
    [26]
    Fei W Y, Fei C. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[DB/OL]. arXiv: 1309.0209v1, 2013.
    [27]
    Wittmuss W. Robust optimization of consumption with random endowment[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2008, 80(5): 459-475.
    [28]
    Wittmuss W. Optimization of dynamic consumption streams under model uncertainty [D]. Germany: Berlin University of Technology, 2010.
    • Supplements(0)
    • Related Articles
    • Track Citations
  • 加载中

Catalog

    |
    Get Citation
    PDF
    XML
    [1]
    Merton R C. Lifetime portfolio selection under uncertainty: The continuous time case[J]. Review of Economics and Statistics, 1969, 51: 247-257.
    [2]
    Merton R C. Optimum consumption and portfolio rules in a continuous time model[J]. Journal of Economic Theory, 1971, 3: 373-413.
    [3]
    Knight F H. Risk, Uncertainty and Profit[M]. Boston: Houghton Mifflin, 1921.
    [4]
    Gilboa I, Schmeidler D. Maxim expected utility with non-unique prior[J]. Journal of Mathematical Economics, 1989, 18:141-153.
    [5]
    Kramkov D, Schachermayer W. The asymptotic elasticity of utility functions and optimal investment in incomplete markets[J]. Annals of Applied Probability, 1999, 9(3): 904-950.
    [6]
    Cvitanic J, Schachermayer W, Wang H. Utility maximization in incomplete markets with random endowment[J]. Finance and Stochastics, 2001, 5(2): 259-272.
    [7]
    Karatzas I, Zitkovic G. Optimal consumption from investment and random endowment in incomplete semimartingale markets[J]. Annals of Probability, 2003, 31(4): 1 821-1 858.
    [8]
    Hugonnier J, Kramkov D. Optimal investment with random endowments in incomplete markets[J]. Annals of Applied Probability, 2004, 14(2): 845-864.
    [9]
    Schied A. Optimal investments for risk- and ambiguity-averse preferences: A duality approach[J]. Finance and Stochastics, 2007, 11(1): 107-129.
    [10]
    Schied A, Wu C T. Duality theory for optimal investments under model uncertainty[J]. Statistics Decisions, 2005, 23(3): 199-217.
    [11]
    Gundel A. Robust utility maximization for complete and incomplete market models[J]. Finance and Stochastics, 2005, 9(2): 151-176.
    [12]
    Duffie D, Zariphopoulou T. Optimal investment with undiversifiable income risk[J]. Mathematical Finance, 1993, 3:135-148.
    [13]
    Quenez M C. Optimal portfolio in a multiple-priors model[J]. Seminar on Stochastic Analysis, Random Fields and Applications Ⅳ, Progress in Probability, 2004, 58: 291-321.
    [14]
    Becherer D. Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging[J]. Annals of Applied Probability, 2006, 16(4): 2 027-2 054.
    [15]
    Muller M. Market completion and robust utility maximization[D]. Berlin: Humboldt University, 2005.
    [16]
    Bauerle N, Rieder U. Portfolio optimization with jumps and unobservable intensity process[J]. Mathematical Finance, 2007, 2(17): 205-224.
    [17]
    Hernandez-Hernandez D, Schied A. A control approach to robust utility maximization with logarithmic utility and time consistent penalties[J]. Stochastic Processes and Applications, 2007, 117(8): 980-1 000.
    [18]
    Fei W Y. Optimal portfolio choice based on α-MEU under ambiguity [J]. Stochastic Models, 2009, 25: 455-482.
    [19]
    Fei W Y. Optimal consumption and portfolio choice with ambiguity and anticipation [J]. Information Sciences, 2007, 177: 5 178-5 190.
    [20]
    Fei Weiyin, Li Shujuan. Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J]. Chinese Journal of Engineering Mathematics, 2012, 29(6): 799-806.
    费为银, 李淑娟. Knight 不确定下带通胀的最优消费和投资模型研究[J]. 工程数学学报, 2012, 29(6): 799-806.
    [21]
    Yang Zhaojun. Maximizing the expected utility from terminal wealth under the case of partial information[J]. Control Theory & Applications, 2006, 20(2): 11-13.
    杨招军. 部分信息下极大化终止时刻期望效用[J]. 控制理论与应用, 2006, 20(2): 11-13.
    [22]
    Xia Dengfeng, Fei Weiyin, Liu Hongjian. On study of optimal investment with ambiguity and anticipation under fluctuated discounting rate[J]. Chinese Journal of Applied Probability and Statistics, 2010, 26(3): 270-276.
    夏登峰, 费为银, 刘宏建. 变折现率下带含糊和预期的投资问题研究[J]. 应用概率统计, 2010, 26(3): 270-276.
    [23]
    Han Liyan, Pan Min. Knightian uncertainty based option pricing with stochastic volatility[J]. Systems Engineering: Theory & Practice, 2012, 32(6): 1 175-1 183.
    韩立岩, 泮敏. 基于奈特不确定性随机波动率期权定价[J]. 系统工程理论与实践, 2012, 32(6): 1 175-1 183.
    [24]
    Li Juan, Fei Weiyin, Shi Xueqin, et al. Optimal trading strategy under disordered asset return and Knightian uncertainty[J]. Applied Mathematics A Journal of Chinese Universities, 2013, 28(1): 13-22.
    李娟, 费为银, 石学芹,等. 奈特不确定下资产收益率发生紊乱的最优投资策略[J]. 高校应用数学学报, 2013, 28(1): 13-22.
    [25]
    Fei Weiyin, Chen Chao, Liang Yong. Optimal consumption-portfolio and retirement problem with disutility under Knightian Uncertainty[J]. Chinese Journal of applied probability and statistics, 2013, 29(1): 53-63.
    费为银, 陈超, 梁勇. Knight 不确定下考虑负效用的消费和投资问题研究[J]. 应用概率统计, 2013, 29(1): 53-63.
    [26]
    Fei W Y, Fei C. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[DB/OL]. arXiv: 1309.0209v1, 2013.
    [27]
    Wittmuss W. Robust optimization of consumption with random endowment[J]. Stochastics: An International Journal of Probability and Stochastic Processes, 2008, 80(5): 459-475.
    [28]
    Wittmuss W. Optimization of dynamic consumption streams under model uncertainty [D]. Germany: Berlin University of Technology, 2010.
    Recommended articles
    TrendMD
    Volume 44 Issue 3 page: 194-202
    Cover

    Article Metrics

    Article views (37) PDF downloads(72)
    Proportional views

    Copyright © Editorial Office of JUSTC, All Rights Reserved. 皖ICP备05002528号

    Supported by: Beijing Renhe Information Technology Co. Ltd    

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return