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May  2021, 17(3): 1451-1469. doi: 10.3934/jimo.2020029

## Utility maximization with habit formation of interaction

 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  May 2021 Early access  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.

Citation: Jingzhen Liu, Yike Wang, Ming Zhou. Utility maximization with habit formation of interaction. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1451-1469. 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. [2] G. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693. [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. [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. [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. [6] J. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283. [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. [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. [9] J. Hicks, Capital and Growth, Oxford Univ. Press, New York, 1965. [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. [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. [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. [13] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer, 1998. doi: 10.1007/b98840. [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. [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. [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. [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. [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. [19] S. Shreve, Stochastic Calculus for Finance, Springer, 2004. [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.

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##### 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. [2] G. Constantinides, Habit formation: A resolution of the equity premium puzzle, Journal of Political Economy, 98 (1990), 519-543.  doi: 10.1086/261693. [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. [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. [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. [6] J. Detemple and F. Zapatero, Asset prices in an exchange economy with habit formation, Econometrica, 59 (1991), 1633-1657.  doi: 10.2307/2938283. [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. [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. [9] J. Hicks, Capital and Growth, Oxford Univ. Press, New York, 1965. [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. [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. [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. [13] I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer, 1998. doi: 10.1007/b98840. [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. [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. [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. [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. [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. [19] S. Shreve, Stochastic Calculus for Finance, Springer, 2004. [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.
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