Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-08T17:32:31.605Z Has data issue: false hasContentIssue false
  • English
  • Français

An optimal consumption and investment problem with partial information

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

Published online by Cambridge University Press:  20 March 2018

Hiroaki Hata  and
Shuenn-Jyi Sheu
Hiroaki Hata*
Affiliation:
Shizuoka University
Shuenn-Jyi Sheu*
Affiliation:
National Central University
*
* Postal address: Department of Mathematics, Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422-852, Japan. Email address: [email protected]
** Postal address: Department of Mathematics, National Central University, Chung-Li, 320-54, Taiwan. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a finite-time optimal consumption problem where an investor wants to maximize the expected hyperbolic absolute risk aversion utility of consumption and terminal wealth. We treat a stochastic factor model in which the mean returns of risky assets depend linearly on underlying economic factors formulated as the solutions of linear stochastic differential equations. We discuss the partial information case in which the investor cannot observe the factor process and uses only past information of risky assets. Then our problem is formulated as a stochastic control problem with partial information. We derive the Hamilton–Jacobi–Bellman equation. We solve this equation to obtain an explicit form of the value function and the optimal strategy for this problem. Moreover, we also introduce the results obtained by the martingale method.

Keywords

Optimal consumption and investmentHARA utilitystochastic factor modelpartial informationHamilton–Jacobi–Bellman equation

MSC classification

Primary: 49L20: Dynamic programming method 60H30: Applications of stochastic analysis (to PDE, etc.) 91G10: Portfolio theory 93E20: Optimal stochastic control
Secondary: 93E11: Filtering
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 131 - 153
Copyright
Copyright © Applied Probability Trust 2018 

References

[1] Bäuerle, N. and Rieder, U. (2007). Portfolio optimization with jumps and unobservable intensity process. Math. Finance. 17, 205–224. CrossRefGoogle Scholar
[2] Bensoussan, A. (1992). Stochastic Control of Partially Observable Systems. Cambridge University Press. Google Scholar
[3] Björk, T., Davis, M. H. A. and Landén, C. (2010). Optimal investment under partial information. Math. Meth. Operat. Res. 71, 371–399. Google Scholar
[4] Brendle, S. (2006). Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723. Google Scholar
[5] Brennan, M. J. (1998). The role of learning in dynamic portfolio decisions. Rev. Finance 1, 295–306. Google Scholar
[6] Bucy, R. S. and Joseph, P. D. (1987). Filtering for Stochastic Processes with Applications to Guidance, 2nd edn. Chelsea, New York. Google Scholar
[7] Callegaro, G., Di Masi, G. B. and Runggaldier, W. J. (2006). Portfolio optimization in discontinuous markets under incomplete information. Asia-Pacific Financial Markets 13, 373–384. CrossRefGoogle Scholar
[8] Castañeda-Leyva, N. and Hernández-Hernández, D. (2005). Optimal consumption-investment problems in incomplete markets with stochastic coefficients. SIAM J. Control Optimization 44, 1322–1344. Google Scholar
[9] Ceci, C. (2013). Optimal investment-consumption for partially observed jump-diffusions. In Seminar on Stochastic Analysis, Random Fields and Applications VII, Birkhäuser, Basel, pp. 337–361. Google Scholar
[10] Chacko, G. and Viceira, L. M. (2005). Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets. Rev. Financial Studies 18, 1369–1402. Google Scholar
[11] Cohen, S. N. and Elliott, R. J. (2015). Stochastic Calculus and Applications, 2nd edn. Springer, Cham. Google Scholar
[12] Cox, J. C. and Huang, C.-F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econom. Theory 49, 33–83. Google Scholar
[13] Cvitanić, J., Lazrak, A., Martellini, L. and Zapatero, F. (2006). Dynamic portfolio choice with parameter uncertainly and the economic value of analysts' recommendations. Rev. Financial Studies 19, 1113–1156. Google Scholar
[14] Detemple, J. B. (1986). Asset pricing in a production economy with incomplete information. J. Finance 41, 383–391. Google Scholar
[15] Dokuchaev, N. (2005). Optimal solution of investment problems via linear parabolic equations generated by Kalman filter. SIAM J. Control Optimization 44, 1239–1258. Google Scholar
[16] Fleming, W. H. and Hernández-Hernández, D. (2003). An optimal consumption model with stochastic volatility. Finance Stoch. 7, 245–262. Google Scholar
[17] Fleming, W. H. and Hernández-Hernández, D. (2005). The tradeoff between consumption and investment in incomplete financial markets. Appl. Math. Optimization 52, 219–235. Google Scholar
[18] Fleming, W. H. and Pang, T. (2004). An application of stochastic control theory to financial economics. SIAM J. Control Optimization 43, 502–531. CrossRefGoogle Scholar
[19] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, Berlin. Google Scholar
[20] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York. Google Scholar
[21] Fouque, J.-P., Papanicolaou, G. and Sircar, K. R. (2000). Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press. Google Scholar
[22] Frey, R., Gabih, A. and Wunderlich, R. (2014). Portfolio optimization under partial information with expert opinions: a dynamic programming approach. Commun. Stoch. Anal. 8, 49–79. Google Scholar
[23] Gennotte, G. (1986). Optimal portfolio choice under incomplete information. J. Finance 41, 733–746. Google Scholar
[24] Hata, H. and Iida, Y. (2006). A risk-sensitive stochastic control approach to an optimal investment problem with partial information. Finance Stoch. 10, 395–426. Google Scholar
[25] Hata, H. and Sheu, S.-J. (2012). On the Hamilton-Jacobi-Bellman equation for an optimal consumption problem: I. Existence of solution. SIAM J. Control Optimization 50, 2373–2400. Google Scholar
[26] Hata, H. and Sheu, S.-J. (2012). On the Hamilton-Jacobi-Bellman equation for an optimal consumption problem: II. Verification theorem. SIAM J. Control Optimization 50, 2401–2430. Google Scholar
[27] Honda, T. (2003). Optimal portfolio choice for unobservable and regime-switching mean returns. J. Econom. Dynam. Control 28, 45–78. Google Scholar
[28] Ibrahim, D. and Abergel, F. (2015). Non-linear filtering and optimal investment under partial information for stochastic volatility models. Preprint. Available at https://arxiv.org/abs/1407.1595. Google Scholar
[29] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York. Google Scholar
[30] Karatzas, I. and Xue, X.-X. (1991). A note on utility maximization under partial observations. Math. Finance 1, 57–70. CrossRefGoogle Scholar
[31] Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a 'small investor' on a finite horizon. SIAM J. Control Optimization 25, 1557–1586. Google Scholar
[32] Karatzas, I., Lehoczky, J. P., Sethi, S. P. and Shreve, S. E. (1986). Explicit solution of a general consumption/investment problem. Math. Operat. Res. 11, 261–294. CrossRefGoogle Scholar
[33] Knox, T. (2003). Learning how to invest when the returns are uncertain. Doctoral thesis. Harvard University. Google Scholar
[34] Lakner, P. (1995). Utility maximization with partial information. Stoch. Process. Appl. 56, 247–273. Google Scholar
[35] Lakner, P. (1998). Optimal trading strategy for an investor: the case of partial information. Stoch. Process. Appl. 76, 77–97. Google Scholar
[36] Lim, T. and Quenez, M.-C. (2015). Portfolio optimization in a default model under full/partial information. Prob. Eng. Inf. Sci. 29, 565–587. CrossRefGoogle Scholar
[37] Lindensjö, K. (2016). Optimal investment and consumption under partial information. Math. Meth. Operat. Res. 83, 87–107. Google Scholar
[38] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes. I. General Theory, 2nd edn. Springer, Berlin. Google Scholar
[39] Liu, J. (2007). Portfolio selection in stochastic environments. Rev. Financial Studies 20, 1–39. Google Scholar
[40] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413. (Erratum: 6 (1973), 213–214.) Google Scholar
[41] Nagai, H. (2001). Risk-sensitive dynamic asset management with partial information. In Stochastics in Finite and Infinite Dimensions, Birkhäuser, Boston, MA, pp. 321–339. Google Scholar
[42] Nagai, H. (2011). Asymptotics of probability minimizing 'down-side' risk under partial information. Quant. Finance 11, 789–803. CrossRefGoogle Scholar
[43] Nagai, H. (2015). H-J-B equations of optimal consumption-investment and verification theorems. Appl. Math. Optimization 71, 279–311. Google Scholar
[44] Nagai, H. and Peng, S. (2002). Risk-sensitive dynamic portfolio optimization with partial information on infinite time horizon. Ann. Appl. Prob. 12, 173–195. Google Scholar
[45] Nagai, H. and Runggaldier, W. J. (2008). PDE approach to utility maximization for market models with hidden Markov factors. In Seminar on Stochastic Analysis, Random Fields and Applications V, Birkhäuser, Basel, pp. 493–506. Google Scholar
[46] Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Math. Operat. Res. 11, 370–382. Google Scholar
[47] Putschögl, W. and Sass, J. (2008). Optimal consumption and investment under partial information. Decis. Econom. Finance 31, 137–170. Google Scholar
[48] Rieder, U. and Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Prob. 42, 362–378. Google Scholar
[49] Rishel, R. (1999). Optimal portfolio management with partial observations and power utility function. In Stochastic Analysis, Control, Optimization and Applications, Birkhäuser, Boston, MA, pp. 605–619. CrossRefGoogle Scholar
[50] Rogers, L. C. G. (2001). The relaxed investor and parameter uncertainty. Finance Stoch. 5, 131–154. Google Scholar
[51] Runggaldier, W. J. and Zaccaria, A. (2000). A stochastic control approach to risk management with restricted information. Math. Finance 10, 277–288. Google Scholar
[52] Sass, J. and Haussmann, U. G. (2004). Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Finance Stoch. 8, 553–577. CrossRefGoogle Scholar
[53] Tamura, T. and Watanabe, Y. (2011). Risk-sensitive portfolio optimization problems for hidden Markov factors on infinite time horizon. Asymptot. Anal. 75, 169–209. Google Scholar
[54] Wachter, J. A. (2002). Portfolio and consumption decisions under mean-reverting returns: an exact solution for complete markets. J. Finance Quant. Anal. 37, 63–91. Google Scholar
[55] Watanabe, Y. (2015). Study of a degenerate elliptic equation in an optimal consumption problem under partial information. Jpn. J. Ind. Appl. Math. 32, 157–185. Google Scholar
[56] Zohar, G. (2001). A generalized Cameron-Martin formula with applications to partially observed dynamic portfolio optimization. Math. Finance 11, 475–494. Google Scholar