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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

 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.
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF01450498. Google Scholar [12] M. Émery, Une topologie sur l'espace des semimartingales,, Séminaire de Probabilités, XIII (1979), 260. Google Scholar [13] H. Föllmer and D. Kramkov, Optional decompositions under constraints,, Probab. Theory Related Fields, 109 (1997), 1. 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. doi: 10.1214/105051604000000134. Google Scholar [17] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.1007/BF00534184. Google Scholar [23] M. Mnif and H. Pham, Stochastic optimization under constraints,, Stochastic Process. Appl., 93 (2001), 149. 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. doi: 10.1214/aoap/1015961159. Google Scholar [25] P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004). Google Scholar [26] R. T. Rockafellar, Extension of Fenchel's duality theorem for convex functions,, Duke Math. J., 33 (1966), 81. 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. 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, 1814 (2003), 95. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF01450498. Google Scholar [12] M. Émery, Une topologie sur l'espace des semimartingales,, Séminaire de Probabilités, XIII (1979), 260. Google Scholar [13] H. Föllmer and D. Kramkov, Optional decompositions under constraints,, Probab. Theory Related Fields, 109 (1997), 1. 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. doi: 10.1214/105051604000000134. Google Scholar [17] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes,, $2^{nd}$ edition, (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. 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. 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. 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. doi: 10.1007/BF00534184. Google Scholar [23] M. Mnif and H. Pham, Stochastic optimization under constraints,, Stochastic Process. Appl., 93 (2001), 149. 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. doi: 10.1214/aoap/1015961159. Google Scholar [25] P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004). Google Scholar [26] R. T. Rockafellar, Extension of Fenchel's duality theorem for convex functions,, Duke Math. J., 33 (1966), 81. 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. 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, 1814 (2003), 95. 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. 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. 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. doi: 10.1090/S0002-9947-1952-0045194-X. Google Scholar
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