American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching
  • MCRF Home
  • This Issue
  • Next Article
    Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation
September  2015, 5(3): 679-695. doi: 10.3934/mcrf.2015.5.679

Constrained nonsmooth utility maximization on the positive real line

Nicholas Westray 1, and  Harry Zheng 2,

1. 

Department of Mathematics, Imperial College, London, SW7 2AZ, United Kingdom
2. 

Department of Mathematics, Imperial College, London SW7 2AZ

Received  March 2014 Revised  December 2014 Published  July 2015

We maximize the expected utility of terminal wealth in an incomplete market where there are cone constraints on the investor's portfolio process and the utility function is not assumed to be strictly concave or differentiable. We establish the existence of the optimal solutions to the primal and dual problems and their dual relationship. We simplify the present proofs in this area and extend the existing duality theory to the constrained nonsmooth setting.
Keywords: convex duality, cone constraints, random endowment., Nonsmooth utility maximization.
Mathematics Subject Classification: Primary: 93E20, 49J52; Secondary: 60H3.
Citation: Nicholas Westray, Harry Zheng. Constrained nonsmooth utility maximization on the positive real line. Mathematical Control & Related Fields, 2015, 5 (3) : 679-695. doi: 10.3934/mcrf.2015.5.679
References:
[1]

F. Bellini and M. Frittelli, On the existence of minimax martingale measures, Math. Finance, 12 (2002), 1-21. doi: 10.1111/1467-9965.00001.  Google Scholar

[2]

S. Biagini and M. Frittelli, Utility maximization in incomplete markets for unbounded processes, Finance Stoch., 9 (2005), 493-517. doi: 10.1007/s00780-005-0163-x.  Google Scholar

[3]

S. Biagini, M. Frittelli and M. Grasselli, Indifference price with general semimartingales, Math. Finance, 21 (2011), 423-446. doi: 10.1111/j.1467-9965.2010.00443.x.  Google Scholar

[4]

B. Bian, S. Miao and H. Zheng, Smooth value functions for a class of nonsmooth utility maximization problems, SIAM J. Financial Math., 2 (2011), 727-747. doi: 10.1137/100793396.  Google Scholar

[5]

B. Bian and H. Zheng, Turnpike property and convergence rate for an investment model with general utility functions, J. Economic Dynamics Control, 51 (2015), 28-49. doi: 10.1016/j.jedc.2014.09.025.  Google Scholar

[6]

B. Bouchard, N. Touzi and A. Zeghal, Dual formulation of the utility maximization problem: The case of nonsmooth utility, Ann. Appl. Probab., 14 (2004), 678-717. doi: 10.1214/105051604000000062.  Google Scholar

[7]

J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM J. Control Optim., 38 (2000), 1050-1066. doi: 10.1137/S036301299834185X.  Google Scholar

[8]

J. Cvitanić, W. Schachermayer and H. Wang, Utility maximization in incomplete markets with random endowment, Finance Stoch., 5 (2001), 259-272. doi: 10.1007/PL00013534.  Google Scholar

[9]

C. Czichowsky, N. Westray and H. Zheng, Convergence in the semimartingale topology and constrained portfolios, Séminaire de Probabilités, 2006 (2011), 395-412. doi: 10.1007/978-3-642-15217-7_17.  Google Scholar

[10]

G. Deelstra, H. Pham and N. Touzi, Dual formulation of the utility maximization problem under transaction costs, Ann. Appl. Probab., 11 (2001), 1353-1383. doi: 10.1214/aoap/1015345406.  Google Scholar

[11]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463-520. doi: 10.1007/BF01450498.  Google Scholar

[12]

M. Émery, Une topologie sur l'espace des semimartingales, Séminaire de Probabilités, XIII (1979), 260-280. Google Scholar

[13]

H. Föllmer and D. Kramkov, Optional decompositions under constraints, Probab. Theory Related Fields, 109 (1997), 1-25. doi: 10.1007/s004400050122.  Google Scholar

[14]

H. G. Heuser, Functional Analysis, Wiley, 1982.  Google Scholar

[15]

E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer, 1965.  Google Scholar

[16]

J. Hugonnier and D. Kramkov, Optimal investment with random endowments in incomplete markets, Ann. Appl. Probab., 14 (2004), 845-864. doi: 10.1214/105051604000000134.  Google Scholar

[17]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, $2^{nd}$ edition, Springer, 2003. doi: 10.1007/978-3-662-05265-5.  Google Scholar

[18]

I. Karatzas and G. Žitković, Optimal consumption from investment and random endowment in incomplete semimartingale markets, Ann. Probab., 31 (2003), 1821-1858. doi: 10.1214/aop/1068646367.  Google Scholar

[19]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), 904-950. doi: 10.1214/aoap/1029962818.  Google Scholar

[20]

D. Kramkov and W. Schachermayer, Necessary and sufficient conditions in the problem of optimal investment in incomplete markets, Ann. Appl. Probab., 13 (2003), 1504-1516. doi: 10.1214/aoap/1069786508.  Google Scholar

[21]

D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1969.  Google Scholar

[22]

J. Mémin, Espaces de semi martingales et changement de probabilité, Z. Wahrsch. Verw. Gebiete, 52 (1980), 9-39. doi: 10.1007/BF00534184.  Google Scholar

[23]

M. Mnif and H. Pham, Stochastic optimization under constraints, Stochastic Process. Appl., 93 (2001), 149-180. doi: 10.1016/S0304-4149(00)00089-2.  Google Scholar

[24]

H. Pham, Minimizing shortfall risk and applications to finance and insurance problems, Ann. Appl. Probab., 12 (2002), 143-172. doi: 10.1214/aoap/1015961159.  Google Scholar

[25]

P. E. Protter, Stochastic Integration and Differential Equations, $2^{nd}$ edition, Springer, 2004.  Google Scholar

[26]

R. T. Rockafellar, Extension of Fenchel's duality theorem for convex functions, Duke Math. J., 33 (1966), 81-89. doi: 10.1215/S0012-7094-66-03312-6.  Google Scholar

[27]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[28]

R. T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math., 39 (1971), 439-469. doi: 10.2140/pjm.1971.39.439.  Google Scholar

[29]

L. C. G. Rogers, Duality in constrained optimal investment and consumption problems: A synthesis, in Paris-Princeton Lectures on Mathematical Finance, Lecture Notes in Math., Springer, 1814 (2003), 95-131. doi: 10.1007/978-3-540-44859-4_3.  Google Scholar

[30]

N. Westray and H. Zheng, Constrained nonsmooth utility maximization without quadratic inf convolution, Stochastic Process. Appl., 119 (2009), 1561-1579. doi: 10.1016/j.spa.2008.08.002.  Google Scholar

[31]

N. Westray and H. Zheng, Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization, Finance Stoch., 15 (2011), 501-512. doi: 10.1007/s00780-010-0128-6.  Google Scholar

[32]

K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc., 72 (1952), 46-66. doi: 10.1090/S0002-9947-1952-0045194-X.  Google Scholar

show all references

References:
[1]

F. Bellini and M. Frittelli, On the existence of minimax martingale measures, Math. Finance, 12 (2002), 1-21. doi: 10.1111/1467-9965.00001.  Google Scholar

[2]

S. Biagini and M. Frittelli, Utility maximization in incomplete markets for unbounded processes, Finance Stoch., 9 (2005), 493-517. doi: 10.1007/s00780-005-0163-x.  Google Scholar

[3]

S. Biagini, M. Frittelli and M. Grasselli, Indifference price with general semimartingales, Math. Finance, 21 (2011), 423-446. doi: 10.1111/j.1467-9965.2010.00443.x.  Google Scholar

[4]

B. Bian, S. Miao and H. Zheng, Smooth value functions for a class of nonsmooth utility maximization problems, SIAM J. Financial Math., 2 (2011), 727-747. doi: 10.1137/100793396.  Google Scholar

[5]

B. Bian and H. Zheng, Turnpike property and convergence rate for an investment model with general utility functions, J. Economic Dynamics Control, 51 (2015), 28-49. doi: 10.1016/j.jedc.2014.09.025.  Google Scholar

[6]

B. Bouchard, N. Touzi and A. Zeghal, Dual formulation of the utility maximization problem: The case of nonsmooth utility, Ann. Appl. Probab., 14 (2004), 678-717. doi: 10.1214/105051604000000062.  Google Scholar

[7]

J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM J. Control Optim., 38 (2000), 1050-1066. doi: 10.1137/S036301299834185X.  Google Scholar

[8]

J. Cvitanić, W. Schachermayer and H. Wang, Utility maximization in incomplete markets with random endowment, Finance Stoch., 5 (2001), 259-272. doi: 10.1007/PL00013534.  Google Scholar

[9]

C. Czichowsky, N. Westray and H. Zheng, Convergence in the semimartingale topology and constrained portfolios, Séminaire de Probabilités, 2006 (2011), 395-412. doi: 10.1007/978-3-642-15217-7_17.  Google Scholar

[10]

G. Deelstra, H. Pham and N. Touzi, Dual formulation of the utility maximization problem under transaction costs, Ann. Appl. Probab., 11 (2001), 1353-1383. doi: 10.1214/aoap/1015345406.  Google Scholar

[11]

F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann., 300 (1994), 463-520. doi: 10.1007/BF01450498.  Google Scholar

[12]

M. Émery, Une topologie sur l'espace des semimartingales, Séminaire de Probabilités, XIII (1979), 260-280. Google Scholar

[13]

H. Föllmer and D. Kramkov, Optional decompositions under constraints, Probab. Theory Related Fields, 109 (1997), 1-25. doi: 10.1007/s004400050122.  Google Scholar

[14]

H. G. Heuser, Functional Analysis, Wiley, 1982.  Google Scholar

[15]

E. Hewitt and K. Stromberg, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer, 1965.  Google Scholar

[16]

J. Hugonnier and D. Kramkov, Optimal investment with random endowments in incomplete markets, Ann. Appl. Probab., 14 (2004), 845-864. doi: 10.1214/105051604000000134.  Google Scholar

[17]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, $2^{nd}$ edition, Springer, 2003. doi: 10.1007/978-3-662-05265-5.  Google Scholar

[18]

I. Karatzas and G. Žitković, Optimal consumption from investment and random endowment in incomplete semimartingale markets, Ann. Probab., 31 (2003), 1821-1858. doi: 10.1214/aop/1068646367.  Google Scholar

[19]

D. Kramkov and W. Schachermayer, The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), 904-950. doi: 10.1214/aoap/1029962818.  Google Scholar

[20]

D. Kramkov and W. Schachermayer, Necessary and sufficient conditions in the problem of optimal investment in incomplete markets, Ann. Appl. Probab., 13 (2003), 1504-1516. doi: 10.1214/aoap/1069786508.  Google Scholar

[21]

D. G. Luenberger, Optimization by Vector Space Methods, Wiley, 1969.  Google Scholar

[22]

J. Mémin, Espaces de semi martingales et changement de probabilité, Z. Wahrsch. Verw. Gebiete, 52 (1980), 9-39. doi: 10.1007/BF00534184.  Google Scholar

[23]

M. Mnif and H. Pham, Stochastic optimization under constraints, Stochastic Process. Appl., 93 (2001), 149-180. doi: 10.1016/S0304-4149(00)00089-2.  Google Scholar

[24]

H. Pham, Minimizing shortfall risk and applications to finance and insurance problems, Ann. Appl. Probab., 12 (2002), 143-172. doi: 10.1214/aoap/1015961159.  Google Scholar

[25]

P. E. Protter, Stochastic Integration and Differential Equations, $2^{nd}$ edition, Springer, 2004.  Google Scholar

[26]

R. T. Rockafellar, Extension of Fenchel's duality theorem for convex functions, Duke Math. J., 33 (1966), 81-89. doi: 10.1215/S0012-7094-66-03312-6.  Google Scholar

[27]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.  Google Scholar

[28]

R. T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math., 39 (1971), 439-469. doi: 10.2140/pjm.1971.39.439.  Google Scholar

[29]

L. C. G. Rogers, Duality in constrained optimal investment and consumption problems: A synthesis, in Paris-Princeton Lectures on Mathematical Finance, Lecture Notes in Math., Springer, 1814 (2003), 95-131. doi: 10.1007/978-3-540-44859-4_3.  Google Scholar

[30]

N. Westray and H. Zheng, Constrained nonsmooth utility maximization without quadratic inf convolution, Stochastic Process. Appl., 119 (2009), 1561-1579. doi: 10.1016/j.spa.2008.08.002.  Google Scholar

[31]

N. Westray and H. Zheng, Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization, Finance Stoch., 15 (2011), 501-512. doi: 10.1007/s00780-010-0128-6.  Google Scholar

[32]

K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc., 72 (1952), 46-66. doi: 10.1090/S0002-9947-1952-0045194-X.  Google Scholar

[1]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033
[2]

Jingzhen Liu, Yike Wang, Ming Zhou. Utility maximization with habit formation of interaction. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1451-1469. doi: 10.3934/jimo.2020029
[3]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174
[4]

Shiyong Li, Wei Sun, Quan-Lin Li. Utility maximization for bandwidth allocation in peer-to-peer file-sharing networks. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1099-1117. doi: 10.3934/jimo.2018194
[5]

Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
[6]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[7]

Yan Zhang, Yonghong Wu, Haixiang Yao. Optimal health insurance with constraints under utility of health, wealth and income. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021031
[8]

Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001
[9]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[10]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[11]

Abdelmalek Aboussoror, Samir Adly, Vincent Jalby. Weak nonlinear bilevel problems: Existence of solutions via reverse convex and convex maximization problems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 559-571. doi: 10.3934/jimo.2011.7.559
[12]

Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial & Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086
[13]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[14]

Sanming Liu, Zhijie Wang, Chongyang Liu. Proximal iterative Gaussian smoothing algorithm for a class of nonsmooth convex minimization problems. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 79-89. doi: 10.3934/naco.2015.5.79
[15]

Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859
[16]

David Yang Gao. Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints. Journal of Industrial & Management Optimization, 2005, 1 (1) : 53-63. doi: 10.3934/jimo.2005.1.53
[17]

Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939
[18]

Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
[19]

Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks & Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223
[20]

Xi-De Zhu, Li-Ping Pang, Gui-Hua Lin. Two approaches for solving mathematical programs with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 951-968. doi: 10.3934/jimo.2015.11.951

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences