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Finite time-horizon optimal investment and consumption with time-varying subsistence consumption constraints. (English) Zbl 1461.91275

Summary: In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time \(t\). Using martingale duality approach and Feynman-Kac formula, we derive the partial differential equation of the Cauchy problem satisfied by the dual value function. We use the integral transform method for solving this Cauchy problem to obtain the general optimal policies in an explicit form. With constant relative risk aversion and constant absolute risk aversion utility functions we illustrate some numerical results of the optimal policies.

MSC:

91G10 Portfolio theory
91G80 Financial applications of other theories
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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[1] Achury, C.; Hubar, S.; Koulovatianos, C., Saving rates and portfolio choice with subsistence consumption, Rev. Econ. Dyn., 15, 108-126 (2012) · doi:10.1016/j.red.2011.01.002
[2] Bertrand, J.; Bertrand, P.; Ovarlez, J., The Mellin Transform, in The Transforms and Applications Handbook (2000), Boca Raton: CRC Press, Boca Raton
[3] Cox, JC; Huang, C., Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory, 49, 33-83 (1989) · Zbl 0678.90011 · doi:10.1016/0022-0531(89)90067-7
[4] Detemple, JB; Zapatero, F., Optimal consumption-portfolio policies with habit formation, Math. Financ., 2, 251-274 (1992) · Zbl 0900.90141 · doi:10.1111/j.1467-9965.1992.tb00032.x
[5] Dybvig, PH, Dusenberry’s ratcheting of consumption: optimal dynamic consumption and investment given intolerance for any decline in standard of living, Rev. Econ. Stud., 62, 287-313 (1995) · Zbl 0830.90025 · doi:10.2307/2297806
[6] Erdlyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, FG, Tables of Integral Transforms (2006), New York: McGraw-Hill, New York
[7] Fleming, W.; Soner, H., Controlled Markov Processes and Viscosity Solutions (2006), New York: Springer, New York · Zbl 1105.60005
[8] Gong, N.; Li, T., Role of index bonds in an optimal dynamic asset allocation model with real subsistence consumption, Appl. Math. Comput., 174, 710-731 (2006) · Zbl 1137.91448
[9] Jeon, JK; Han, HJ; Kang, MJ, Valuing American floating strike lookback option and neumann problem for inhomogeneous Black-Scholes equation, J. Comput. Appl. Math., 313, 218-234 (2017) · Zbl 1354.35162 · doi:10.1016/j.cam.2016.09.020
[10] Jeon, JK; Koo, HK; Shin, YH, Portfolio selection with consumption ratcheting, J. Econ. Dyn. Control, 92, 153-182 (2018) · Zbl 1401.91520 · doi:10.1016/j.jedc.2018.05.003
[11] Karatzas, I.; Lehoczky, JP; Sethi, SP; Shreve, SE, Explicit solution of a general consumption/investment problem, Math. Operations Res., 11, 261-294 (1986) · doi:10.1287/moor.11.2.261
[12] Karatzas, I.; Lehoczky, JP; Shreve, SE, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim., 25, 1557-1586 (1987) · Zbl 0644.93066 · doi:10.1137/0325086
[13] Karatzas, I.; Shreve, SE, Methods of Mathematical Finance (1998), New York: Springer, New York · Zbl 0941.91032 · doi:10.1007/978-1-4939-6845-9
[14] Lakner, P.; Nygren, LM, Portfolio optimization with downside constraints, Math. Financ., 16, 283-299 (2006) · Zbl 1145.91350 · doi:10.1111/j.1467-9965.2006.00272.x
[15] Lio, M.; Arrow, KJ; Ng, YK; Yang, X., The inframarginal analysis of demand and supply and the relationship between a minimum level of consumption and the division of labour, Increasing Returns and Economic Analysis (1998), London: Palgrave Macmillan, London
[16] Merton, RC, Lifetime portfolio selection under uncertainty: the continuous-time case, Rev. Econ. Stat., 51, 247-257 (1969) · doi:10.2307/1926560
[17] Merton, RC, Optimum consumption and portfolio rules in a continuous-time model, J. Econ. Theory, 3, 373-413 (1971) · Zbl 1011.91502 · doi:10.1016/0022-0531(71)90038-X
[18] Panini, R.; Srivastav, RP, Option pricing with mellin transforms, Math. Comput. Model., 40, 43-56 (2004) · Zbl 1112.91037 · doi:10.1016/j.mcm.2004.07.008
[19] Panini, R.; Srivastav, RP, Pricing perpetual options using Mellin transforms, Appl. Math. Lett., 18, 471-474 (2005) · Zbl 1111.91018 · doi:10.1016/j.aml.2004.03.012
[20] Shim, G., Shin,Y.H.: Portfolio selection with subsistence consumption constraints and CARA utility. Math. Probl. Eng. 6 (2014) · Zbl 1407.91236
[21] Shin, YH; Lim, BH; Choi, UJ, Optimal consumption and portfolio selection problem with downside consumption constraints, Appl. Math. Comput., 188, 1801-1811 (2007) · Zbl 1298.91151
[22] Sneddon, IN, The Use of Integral Transforms (1972), New York: McGraw-Hill, New York · Zbl 0237.44001
[23] Yuan, H.; Hu, Y., Optimal consumption and portfolio policies with the consumption habit constraints and the terminal wealth downside constraints, Insur. Math. Econ., 45, 405-409 (2009) · Zbl 1231.91418 · doi:10.1016/j.insmatheco.2009.08.012
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