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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[25]

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

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

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