American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2020029

Utility maximization with habit formation of interaction

Jingzhen Liu 1, , Yike Wang 2,, and  Ming Zhou 2,

1. 

Department of Actuarial Science, School of Insurance, , Central University of Finance and Economics Beijing 100081, China
2. 

Department of Actuarial Science, School of Insurance; , China Institute for Actuarial Science, , Central University of Finance and Economics, Beijing 100081, China

* Corresponding author: Yike Wang

Received  February 2019 Revised  September 2019 Published  February 2020

Fund Project: The first author is supported by Projects 11771466 and 11571388 supported by National Natural Science Foundation of China, and Program for Innovation Research in Central University of Finance and Economics

In this paper, we analytically solve the utility maximization problem for a consumption set with multiple habit formation of interaction. Consumption is here composed of habitual and nonhabitual components, where habitual consumption represents the effect of past consumption. We further assume that the individual seeks to maximize his/her expected utility from nonhabitual consumption. Although there is usually no explicit solution of linear dynamic systems in the habit formation model, we study the functional minimum of habitual consumption. To solve the optimization problem with a general utility function, we adopt the convex dual martingale approach to construct the optimal consumption strategy and provide an economic interpretation for nearly every object throughout the solution process.

Keywords: consumption-portfolio optimization, habit formation, dynamic system, martingale approach, convex dual.
Mathematics Subject Classification: Primary: 91B42, 91G80; Secondary: 93E03.
Citation: Jingzhen Liu, Yike Wang, Ming Zhou. Utility maximization with habit formation of interaction. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020029
References:
[1]

J. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.  Google Scholar

[2]

G. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693.  Google Scholar

[3]

J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767-818.  doi: 10.1214/aoap/1177005576.  Google Scholar

[4]

J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652-681.  doi: 10.1214/aoap/1177005357.  Google Scholar

[5]

J. Detemple and I. Karatzas, Non-addictive habits: Optimal consumption portfolio policies, Journal of Economic Theory, 113 (2003), 265-285.  doi: 10.1016/S0022-0531(03)00099-1.  Google Scholar

[6]

J. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283.  Google Scholar

[7]

J. Detemple and F. Zapatero, Optimal consumption-portfolio policies with habit formation, Mathematical Finance, 2 (1992), 251-274.  doi: 10.1111/j.1467-9965.1992.tb00032.x.  Google Scholar

[8]

N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic pdes, SIAM Journal on Control and Optimization, 48 (2009), 481-520.  doi: 10.1137/070686998.  Google Scholar

[9] J. Hicks, Capital and Growth, Oxford Univ. Press, New York, 1965.   Google Scholar
[10]

J. Kakeu and P. Nguimkeu, Habit formation and exhaustible resource risk-pricing, Energy Economics, 64 (2017), 1-12.  doi: 10.1016/j.eneco.2017.03.013.  Google Scholar

[11]

I. Karatzas, J. Lehoczky, S. Sethi and S. Shreve, Explicit solution of a general consumption/investment problem, Mathematics of Operations Research, 11 (1986), 261-294.  doi: 10.1287/moor.11.2.261.  Google Scholar

[12]

I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a "small investor" on a finite horizon, SIAM Journal on Control and Optimization, 25 (1987), 1557-1586.  doi: 10.1137/0325086.  Google Scholar

[13]

I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer, 1998. doi: 10.1007/b98840.  Google Scholar

[14]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560.  Google Scholar

[15]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[16]

C. Munk, Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences, Journal of Economic Dynamics and Control, 32 (2008), 3560-3589.  doi: 10.1016/j.jedc.2008.02.005.  Google Scholar

[17]

R. Muraviev, Additive habit formation: consumption in incomplete markets with random endowments, Mathematics and Financial Economics, 5 (2011), 67-99.  doi: 10.1007/s11579-011-0049-y.  Google Scholar

[18]

M. Schroder and C. Skiadas, An isomorphism between asset pricing models with and without linear habit formation, Review of Financial Studies, 15 (2002), 1189-1221.   Google Scholar

[19]

S. Shreve, Stochastic Calculus for Finance, Springer, 2004.  Google Scholar

[20]

S. Sundaresan, Intertemporally dependent preferences and the volatility of consumption and wealth, Review of Financial Studies, 2 (1989), 73-89.  doi: 10.1093/rfs/2.1.73.  Google Scholar

show all references

References:
[1]

J. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.  Google Scholar

[2]

G. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693.  Google Scholar

[3]

J. Cvitanić and I. Karatzas, Convex duality in constrained portfolio optimization, Annals of Applied Probability, 2 (1992), 767-818.  doi: 10.1214/aoap/1177005576.  Google Scholar

[4]

J. Cvitanić and I. Karatzas, Hedging contingent claims with constrained portfolios, Annals of Applied Probability, 3 (1993), 652-681.  doi: 10.1214/aoap/1177005357.  Google Scholar

[5]

J. Detemple and I. Karatzas, Non-addictive habits: Optimal consumption portfolio policies, Journal of Economic Theory, 113 (2003), 265-285.  doi: 10.1016/S0022-0531(03)00099-1.  Google Scholar

[6]

J. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283.  Google Scholar

[7]

J. Detemple and F. Zapatero, Optimal consumption-portfolio policies with habit formation, Mathematical Finance, 2 (1992), 251-274.  doi: 10.1111/j.1467-9965.1992.tb00032.x.  Google Scholar

[8]

N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic pdes, SIAM Journal on Control and Optimization, 48 (2009), 481-520.  doi: 10.1137/070686998.  Google Scholar

[9] J. Hicks, Capital and Growth, Oxford Univ. Press, New York, 1965.   Google Scholar
[10]

J. Kakeu and P. Nguimkeu, Habit formation and exhaustible resource risk-pricing, Energy Economics, 64 (2017), 1-12.  doi: 10.1016/j.eneco.2017.03.013.  Google Scholar

[11]

I. Karatzas, J. Lehoczky, S. Sethi and S. Shreve, Explicit solution of a general consumption/investment problem, Mathematics of Operations Research, 11 (1986), 261-294.  doi: 10.1287/moor.11.2.261.  Google Scholar

[12]

I. Karatzas, J. Lehoczky and S. Shreve, Optimal portfolio and consumption decisions for a "small investor" on a finite horizon, SIAM Journal on Control and Optimization, 25 (1987), 1557-1586.  doi: 10.1137/0325086.  Google Scholar

[13]

I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer, 1998. doi: 10.1007/b98840.  Google Scholar

[14]

R. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.  doi: 10.2307/1926560.  Google Scholar

[15]

R. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar

[16]

C. Munk, Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences, Journal of Economic Dynamics and Control, 32 (2008), 3560-3589.  doi: 10.1016/j.jedc.2008.02.005.  Google Scholar

[17]

R. Muraviev, Additive habit formation: consumption in incomplete markets with random endowments, Mathematics and Financial Economics, 5 (2011), 67-99.  doi: 10.1007/s11579-011-0049-y.  Google Scholar

[18]

M. Schroder and C. Skiadas, An isomorphism between asset pricing models with and without linear habit formation, Review of Financial Studies, 15 (2002), 1189-1221.   Google Scholar

[19]

S. Shreve, Stochastic Calculus for Finance, Springer, 2004.  Google Scholar

[20]

S. Sundaresan, Intertemporally dependent preferences and the volatility of consumption and wealth, Review of Financial Studies, 2 (1989), 73-89.  doi: 10.1093/rfs/2.1.73.  Google Scholar

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