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Optimal Portfolio-Consumption with Habit Formation under Partial Observations Open Access


Other title
Habit formation
partial observation
stochastic differential equation
Type of item
Degree grantor
University of Alberta
Author or creator
Huang, Hanlin
Supervisor and department
Choulli, Tahir (Mathematical and Statistical Sciences)
Examining committee member and department
Choulli, Tahir (Mathematical and Statistical Sciences)
Frei, Christoph (Mathematical and Statistical Sciences)
Melnikov, Alexander (Mathematical and Statistical Sciences)
Minev, Peter (Mathematical and Statistical Sciences)
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Master of Science
Degree level
The aim of my thesis consists of characterizing explicitly the optimal consumption and investment strategy for an investor, when her habit level process is incorporated in the utility formulation. For a continuous-time market model, I maximize the expected utility from terminal wealth and/or consumption. For this optimization problem, the thesis presents three novel contributions. Using the Kalman-Bucy filter theorem, I transform the optimization problem under the partial information into an equivalent optimization problem within a full information context. Using the stochastic control techniques, this latter problem is reduced to solve an associated Hamilton-Jacobi-Bellman equation (HJB hereafter). For the exponential utility, the solution to the HJB is explicitly described, while the optimal policies/controls as well as the optimal wealth process are described by a stochastic differential equation. Furthermore, I discuss qualitative analysis on the optimal policies for the exponential utility. These achievements constitute my first contribution in this thesis. The second contribution lies in considering a stochastic volatility model and addressing the same optimization problem using again the techniques of stochastic control. The third contribution of my thesis resides in combining the filtering techniques with the martingale approach to solve the optimization problem when the investor is endowed with the logarithm, power or exponential utility.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
Abel, A. (1990): Asset Prices under Habit Formation and Catching up with the Joneses, American Economic Review, 80 38–42Björk, T., Davis, H A M. and Landén, C. (2010): Optimal investment under partial information, Mathematical Methods of Operations Re- search,74, 2 371–399Brendle, S. (2006): Portfolio selection under incomplete information, Stochastic Processes and their Applications 116 701–723Brennan, M.J. and Xia, Y. (2001): Assessing assent pricing anomalies, Review of Financial Studies, 14 905–942Browne, S and Whitt, D. (1996):Portfolio Choice and the Bayesian Kelly Criterion, Advances in Applied Probability, 28, 4 1145-1176Chisholm, J. S. R. (2006): Continued fraction solution of the general Riccati equation, Rational Approximation and Interpolation, Springer, 1105 109-116Choulli, T. and Stricker, C. (2009): Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure, Stochastic Processes and their Applications, 119 1368–1385Detemple, J B. and Zapatero, F. (1992): Optimal Consumption-Portfolio Policies With Habit Formation, Mathematical Finance, 2, 4 251–274Detemple, J B. and Zapatero, F. (1991): Asset Prices in an Exchange Economy with Habit Formation, Econometrica, 59 1633–1657Fisher, I. (1930): The theory of interest, The Macmillan Company, New YorkHe, S., Wang, J. and Yan, J. (1992): Semimartingale theory and stochas- tic calculus, Science Press and CRC PressIbrahim, D. and Abergel, F. (2014): Non linear filtering and optimal investment under partial information for stochastic volatility models,
, R. E. and Bucy, R. S. (1961): New Results in Linear Filtering and Prediction Theory, Trans. ASME Ser. D. J. Basic Engineering, 83 95–108Karatzas, I., Lehoczky,J. P. and Shreve,S. E. (1991): Optimal Portfolio and Consumption Decisions for a "Small Investor" on a Finite Horizon, SIAM J. Control and optimization, 25 1557–1586Korn, R. and Korn, E (2001): Option pricing and portfolio optimization: Modern methods of financial mathematics, Graduate Studies in Mathematics, American Mathematical Society, 31Lakner, P. (1995): Utility maximization with partial information, Stochastic Processes and their Applications, 56 247–273Mehra, R. and Prescott, E. C. (1985): The equity premium: A puzzle, Journal of Monetary Economics, 15,2 145–161Monoyios, M. (2009): Optimal investment and hedging under partial and inside information, Advanced Financial Modeling (edited by Hansjörg Albrecher, Wolfgang J. Runggaldier, Walter Schachermayer), Radon Series Comp. Appl. Math 8 371–410Sundaresan, S M. (1989): Intertemporally dependent preferences and the volatility of consumption and wealth, Review of Financial Studies, 2, 1 73–89Xiong, J. (1997): An Introduction to Stochastic Filtering Theory, Oxford Mathematics PressYong, J. and Zhou, X. (1999): Stochastic Controls – Hamiltonian Systems and HJB Equations, SpringerYu, X. (2014): An explicit example of optimal portfolio choices with habit formation and partial information,Annals of Applied Probability 25, 3 1383–1419Yu, X. (2012): Utility maximization with consumption habit formation in incomplete markets. Ph.D. Thesis, The University of Texas at Austin

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