Optimal consumption and investment with power utility

In this thesis we study the utility maximization problem for power utility random elds in a general semimartingale nancial market, with and without intermediate consumption. The notion of an opportunity process is introduced as a reduced form of the value process for the resulting stochastic control problem. This process is shown to describe the key objects: the optimal strategy, the value function, and the convex-dual problem. We show that the existence of an optimal strategy implies that the opportunity process solves the so-called Bellman equation. The optimal strategy is described pointwise in terms of the opportunity process, which is also characterized as the minimal solution of the Bellman equation. Furthermore, we provide veri cation theorems for this equation. As an example, we consider exponential Lévy models, for which we construct an explicit solution in terms of the Lévy triplet. Finally, we study the asymptotic properties of the optimal strategy as the relative risk aversion tends to in nity or to one. The convergence of the optimal consumption is obtained for the general case, while the convergence of the optimal trading strategy is obtained for continuous semimartingale models.

[1]  Marcel Nutz.  Risk aversion asymptotics for power utility maximization , 2010, 1003.3582.

[2]  Christoph Czichowsky,et al.  Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands , 2011 .

[3]  Marcel Nutz,et al.  The opportunity process for optimal consumption and investment with power utility , 2009, 0912.1879.

[4]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[5]  M. Schweizer.  Minimal Entropy Martingale Measure , 2010 .

[6]  Marcel Nutz,et al.  The Bellman equation for power utility maximization with semimartingales , 2009, ArXiv.

[7]  Marcel Nutz.  Power Utility Maximization in Constrained Exponential Lévy Models , 2009 .

[8]  Tahir Choulli,et al.  Comparing the minimal Hellinger martingale measure of order q to the q-optimal martingale measure , 2009 .

[9]  Kasper Larsen.  Continuity of Utility-Maximization with Respect to Preferences , 2009 .

[10]  Constantinos Kardaras,et al.  NO‐FREE‐LUNCH EQUIVALENCES FOR EXPONENTIAL LÉVY MODELS UNDER CONVEX CONSTRAINTS ON INVESTMENT , 2008, 0803.2169.

[11]  F. Delbaen,et al.  Optimality and risk : modern trends in mathematical finance : the Kabanov festschrift , 2009 .

[12]  Martin Schweizer,et al.  Exponential Utility Indifference Valuation in a General Semimartingale Model , 2009 .

[13]  Johannes Muhle-Karbe,et al.  On Utility-Based Investment, Pricing and Hedging in Incomplete Markets , 2009 .

[14]  Christian Bender,et al.  On q-optimal martingale measures in exponential Lévy models , 2008, Finance Stochastics.

[15]  Ying Hu,et al.  Quadratic BSDEs with convex generators and unbounded terminal conditions , 2007, math/0703423.

[16]  Monique Jeanblanc,et al.  MINIMAL f q -MARTINGALE MEASURES FOR EXPONENTIAL LÉVY PROCESSES , 2007, 0710.5594.

[17]  Constantinos Kardaras,et al.  The numéraire portfolio in semimartingale financial models , 2007, Finance Stochastics.

[18]  C. Kardaras,et al.  STABILITY OF THE UTILITY MAXIMIZATION PROBLEM WITH RANDOM ENDOWMENT IN INCOMPLETE MARKETS , 2007, 0706.0482.

[19]  P. Grandits,et al.  RISK AVERSE ASYMPTOTICS AND THE OPTIONAL DECOMPOSITION , 2007 .

[20]  Sara Biagini,et al.  The supermartingale property of the optimal wealth process for general semimartingales , 2007, Finance Stochastics.

[21]  Dirk Becherer.  Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging , 2006 .

[22]  Miklós Rásonyi,et al.  Convergence of Utility Indifference Prices to the Superreplication Price , 2006, Math. Methods Oper. Res..

[23]  S. Dreyfus,et al.  Lifetime Portfolio Selection under Uncertainty: the Continuous-time Case , 2006 .

[24]  Sasha Stoikov,et al.  Dynamic Asset Allocation and Consumption Choice in Incomplete Markets , 2005 .

[25]  Sara Biagini,et al.  Utility maximization in incomplete markets for unbounded processes , 2005, Finance Stochastics.

[26]  P. Imkeller,et al.  Utility maximization in incomplete markets , 2005, math/0508448.

[27]  M. Schweizer,et al.  Dynamic exponential utility indifference valuation , 2005, math/0508489.

[28]  J. Kallsen,et al.  On the Structure of General Mean-Variance Hedging Strategies , 2005, 0708.1715.

[29]  Marina Santacroce.  On the Convergence of the p-Optimal Martingale Measures to the Minimal Entropy Martingale Measure , 2005 .

[30]  Elyès Jouini,et al.  Convergence of utility functions and convergence of optimal strategies , 2004, Finance Stochastics.

[31]  Michael R. Tehranchi.  Explicit solutions of some utility maximization problems in incomplete markets , 2004 .

[32]  J. Kallsen,et al.  σ-LOCALIZATION AND σ-MARTINGALES∗ , 2004 .

[33]  Martino Grasselli.  A stability result for the HARA class with stochastic interest rates , 2003 .

[34]  W. Schachermayer,et al.  Necessary and sufficient conditions in the problem of optimal investment in incomplete markets , 2003 .

[35]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[36]  J. Kallsen,et al.  A Complete Explicit Solution to the Log-Optimal Portfolio Problem , 2003 .

[37]  Gordan Žitković,et al.  A Filtered Version of the Bipolar Theorem of Brannath and Schachermayer , 2007, 0706.0049.

[38]  Y. Kabanov,et al.  On the optimal portfolio for the exponential utility maximization: remarks to the six‐author paper , 2002 .

[39]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[40]  Kenneth H. Karlsen,et al.  Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution , 2001, Finance Stochastics.

[41]  Ludger Rüschendorf,et al.  Minimax and minimal distance martingale measures and their relationship to portfolio optimization , 2001, Finance Stochastics.

[42]  I. Karatzas,et al.  Optimal Consumption from Investment and Random Endowment in Incomplete Semimartingale Markets , 2001, 0706.0051.

[43]  W. Schachermayer.  Optimal investment in incomplete markets when wealth may become negative , 2001 .

[44]  Thaleia Zariphopoulou,et al.  A solution approach to valuation with unhedgeable risks , 2001, Finance Stochastics.

[45]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[46]  C. Gollier.  The economics of risk and time , 2001 .

[47]  Jan Kallsen,et al.  Optimal portfolios for logarithmic utility , 2000 .

[48]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[49]  M. Kobylanski.  Backward stochastic differential equations and partial differential equations with quadratic growth , 2000 .

[50]  P. Grandits.  On Martingale Measures for Stochastic Processes with Independent Increments , 2000 .

[51]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[52]  W. Schachermayer,et al.  The asymptotic elasticity of utility functions and optimal investment in incomplete markets , 1999 .

[53]  Huyên Pham,et al.  Dynamic programming and mean-variance hedging , 1999, Finance Stochastics.

[54]  F. Delbaen,et al.  The fundamental theorem of asset pricing for unbounded stochastic processes , 1998 .

[55]  Walter Schachermayer,et al.  Weighted norm inequalities and hedging in incomplete markets , 1997, Finance Stochastics.

[56]  Ts Kim,et al.  Dynamic Nonmyopic Portfolio Behavior , 1994 .

[57]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[58]  M. Schweizer.  On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition , 1995 .

[59]  Kim C. Border,et al.  Infinite Dimensional Analysis: A Hitchhiker’s Guide , 1994 .

[60]  Christophe Stricker,et al.  Décomposition de Kunita-Watanabe , 1993 .

[61]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[62]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[63]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[64]  P. Samuelson.  Why we should not make mean log of wealth big though years to act are long , 1979 .

[65]  M. Émery.  Une Topologie sur l’Espace des Semimartingales , 1979 .

[66]  C. Doléans-Dade,et al.  Inegalites de normes avec poids , 1979 .

[67]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[68]  R. Rockafellar,et al.  Integral functionals, normal integrands and measurable selections , 1976 .

[69]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[70]  P. Samuelson.  LIFETIME PORTFOLIO SELECTION BY DYNAMIC STOCHASTIC PROGRAMMING , 1969 .

[71]  J. Mossin.  Optimal multiperiod portfolio policies , 1968 .