Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T23:20:57.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal entry and consumption under habit formation

Part of: Stochastic systems and control Mathematical finance Hamilton-Jacobi theories, including dynamic programming

Published online by Cambridge University Press:  10 March 2022

Yue Yang  and
Xiang Yu Open the ORCID record for Xiang Yu [Opens in a new window]
Yue Yang*
Affiliation:
The Hong Kong Polytechnic University
Xiang Yu*
Affiliation:
The Hong Kong Polytechnic University
*
*Postal address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
*Postal address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies a composite problem involving decision-making about the optimal entry time and dynamic consumption afterwards. In Stage 1, the investor has access to full market information subject to some information costs and needs to choose an optimal stopping time to initiate Stage 2; in Stage 2, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observation of free public stock prices. Habit formation preferences are employed, in which past consumption affects the investor’s current decisions. Using the stochastic Perron method, the value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

Keywords

Optimal entry problemconsumption habit formationstochastic Perron methodviscosity solution

MSC classification

Primary: 91G10: Portfolio theory 49L25: Viscosity solutions
Secondary: 93E11: Filtering 93E20: Optimal stochastic control 49L20: Dynamic programming method
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 433 - 459
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahearne, A. G., Griever, W. L. and Warnock, F. E. (2004). Information costs and home bias: an analysis of US holdings of foreign equities. J. Internat. Econom. 62, 313–336.CrossRefGoogle Scholar
Bayraktar, E. and Sirbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: the linear case. Proc. Amer. Math. Soc. 140, 3645–3654.CrossRefGoogle Scholar
Bayraktar, E. and Sirbu, M. (2013). Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optimization 51, 4274–4294.CrossRefGoogle Scholar
Bayraktar, E. and Sirbu, M. (2014). Stochastic Perron’s method and verification without smoothness using viscosity comparison: obstacle problems and Dynkin games. Proc. Amer. Math. Soc. 142, 1399–1412.CrossRefGoogle Scholar
Bayraktar, E. and Zhang, Y. (2015). Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. SIAM J. Control Optimization 53, 91–113.CrossRefGoogle Scholar
Björk, T., Davis, M. and Landén, C. (2010). Optimal investment under partial information. Math. Meth. Operat. Res. 71, 371–399.CrossRefGoogle Scholar
Bo, L., Liao, H. and Yu, X. (2019). Risk-sensitive credit portfolio optimization under partial information and contagion risk. Preprint. Available at https://arxiv.org/abs/1905.08004.Google Scholar
Brendle, S. (2006). Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723.CrossRefGoogle Scholar
Brennan, M. J. and Xia, Y. (2010). Persistence, predictability, and portfolio planning. In Handbook of Quantitative Finance and Risk Management, Springer, Boston, pp. 289–318.CrossRefGoogle Scholar
Campbell, J. Y. et al. (1997). The Econometrics of Financial Markets. Princeton University Press.CrossRefGoogle Scholar
Constantinides, G. M. (1990). Habit formation: a resolution of the equity premium puzzle. J. Political Econom. 98, 519–543.CrossRefGoogle Scholar
Detemple, J. and Zapatero, F. (1992). Optimal consumption-portfolio policies with habit formation. Math. Finance 2, 251–274.CrossRefGoogle Scholar
Duckworth, J. K. and Zervos, M. (2000). An investment model with entry and exit decisions. J. Appl. Prob. 37, 547–559.CrossRefGoogle Scholar
Englezos, N. and Karatzas, I. (2009). Utility maximization with habit formation: dynamic programming and stochastic PDEs. SIAM J. Control Optimization 48, 481–520.CrossRefGoogle Scholar
Fama, E. F. and French, K. R. (1989). Business conditions and expected returns on stocks and bonds. J. Financial Econom. 25, 23–49.CrossRefGoogle Scholar
Friedman, A. (2012). Stochastic Differential Equations and Applications. Dover, Mineola, NY.Google Scholar
Janeček, K. and Sîrbu, M. (2012). Optimal investment with high-watermark performance fee. SIAM J. Control Optimization 50, 790–819.CrossRefGoogle Scholar
Kang, J. and Stulz, R. M. (1997). Why is there a home bias? An analysis of foreign portfolio equity ownership in Japan. J. Financial Econom. 46, 3–28.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Keppo, J., Tan, H. M. and Zhou, C. (2019). Smart city investments. Preprint. Available at https://doi.org/10.2139/ssrn.3141043.CrossRefGoogle Scholar
Kim, T. S. and Omberg, E. (1996). Dynamic nonmyopic portfolio behavior. Rev. Financial Studies 9, 141–161.CrossRefGoogle Scholar
Lakner, P. (1998). Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97.CrossRefGoogle Scholar
Lee, J., Yu, X. and Zhou, C. (2021). Lifetime ruin under high-water mark fees and drift uncertainty. Appl. Math. Optimization 84, 2743–2773.CrossRefGoogle Scholar
Mehra, R. and Prescott, E. C. (1985). The equity premium: a puzzle. J. Monetary Econom. 15, 145–161.CrossRefGoogle Scholar
Monoyios, M. (2009). Optimal investment and hedging under partial and inside information. Adv. Financial Model. 8, 371–410.Google Scholar
Munk, C. (2008). Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences. J. Econom. Dynam. Control 32, 3560–3589.CrossRefGoogle Scholar
Pham, H. (1997). Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optimization 35, 145–164.CrossRefGoogle Scholar
Pham, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Portes, R. and Rey, H. (2005). The determinants of cross-border equity flows. J. Internat. Econom. 65, 269–296.CrossRefGoogle Scholar
Poterba, J. M. and Summers, L. H. (1988). Mean reversion in stock prices: evidence and implications. J. Financial Econom. 22, 27–59.CrossRefGoogle Scholar
Reikvam, K. (1998). Viscosity solutions of optimal stopping problems. Stoch. Stoch. Reports 62, 285–301.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Sirbu, M. (2014). Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optimization 52, 1693–1711.CrossRefGoogle Scholar
Xia, Y. (2001). Learning about predictability: the effects of parameter uncertainty on dynamic asset allocation. J. Finance 56, 205–246.CrossRefGoogle Scholar
Yu, X. (2015). Utility maximization with addictive consumption habit formation in incomplete semimartingale markets. Ann. Appl. Prob. 25, 1383–1419.CrossRefGoogle Scholar
Yu, X. (2017). Optimal consumption under habit formation in markets with transaction costs and random endowments. Ann. Appl. Prob. 27, 960–1002.CrossRefGoogle Scholar