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Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets,, SIAM J. Control Optim., 38 (2000), 1050.
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.
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.
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.
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.
doi: 10.1007/BF01450498. |
[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. |
[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.
doi: 10.1214/105051604000000134. |
[17] |
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes,, $2^{nd}$ edition, (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.
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.
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.
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.
doi: 10.1007/BF00534184. |
[23] |
M. Mnif and H. Pham, Stochastic optimization under constraints,, Stochastic Process. Appl., 93 (2001), 149.
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.
doi: 10.1214/aoap/1015961159. |
[25] |
P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004).
|
[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. |
[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.
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, 1814 (2003), 95.
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.
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.
doi: 10.1007/s00780-010-0128-6. |
[32] |
K. Yosida and E. Hewitt, Finitely additive measures,, Trans. Amer. Math. Soc., 72 (1952), 46.
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.
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.
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.
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.
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.
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.
doi: 10.1214/105051604000000062. |
[7] |
J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets,, SIAM J. Control Optim., 38 (2000), 1050.
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.
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.
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.
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.
doi: 10.1007/BF01450498. |
[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. |
[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.
doi: 10.1214/105051604000000134. |
[17] |
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes,, $2^{nd}$ edition, (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.
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.
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.
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.
doi: 10.1007/BF00534184. |
[23] |
M. Mnif and H. Pham, Stochastic optimization under constraints,, Stochastic Process. Appl., 93 (2001), 149.
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.
doi: 10.1214/aoap/1015961159. |
[25] |
P. E. Protter, Stochastic Integration and Differential Equations,, $2^{nd}$ edition, (2004).
|
[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. |
[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.
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, 1814 (2003), 95.
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.
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.
doi: 10.1007/s00780-010-0128-6. |
[32] |
K. Yosida and E. Hewitt, Finitely additive measures,, Trans. Amer. Math. Soc., 72 (1952), 46.
doi: 10.1090/S0002-9947-1952-0045194-X. |
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