Smooth Ambiguity Aversion toward Small Risks and Continuous-Time Recursive Utility

Assuming Brownian/Poisson uncertainty, a certainty equivalent (CE) based on the smooth second-order expected utility of Klibanoff, Marinacci, and Mukerji is shown to be approximately equal to an expected-utility CE. As a consequence, the corresponding continuous-time recursive utility form is the same as for Kreps-Porteus utility. The analogous conclusions are drawn for a smooth divergence CE, based on a formulation of Maccheroni, Marinacci, and Rustichini, but only under Brownian uncertainty. Under Poisson uncertainty, a smooth divergence CE can be approximated with an expected-utility CE if and only if it is of the entropic type. A nonentropic divergence CE results in a new class of continuous-time recursive utilities that price Brownian and Poissonian risks differently.

[1]  M. Machina.  "Expected Utility" Analysis without the Independence Axiom , 1982 .

[2]  É. Pardoux,et al.  Probabilistic interpretation of a system of semi-linear parabolic partial differential equations , 1997 .

[3]  Paul R. Milgrom,et al.  AGGREGATION AND LINEARITY IN THE PROVISION OF INTERTEMPORAL INCENTIVES , 1987 .

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

[5]  Costis Skiadas,et al.  Asset Pricing Theory , 2009 .

[6]  R. Barro.  Rare Disasters and Asset Markets in the Twentieth Century , 2006 .

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

[8]  Massimo Marinacci,et al.  Dynamic variational preferences , 2006, J. Econ. Theory.

[9]  Uzi Segal,et al.  The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach , 1987 .

[10]  X. Gabaix.  Variable Rare Disasters: A Tractable Theory of Ten Puzzles in Macro-finance , 2008 .

[11]  Daniel E. Geer,et al.  Risk Aversion , 2012, IEEE Secur. Priv..

[12]  John B. Donaldson,et al.  The Structure of Intertemporal Preferences under Uncertainty and Time Consistent Plans , 1985 .

[13]  H. P..  Annales de l'Institut Henri Poincaré , 1931, Nature.

[14]  Lars Peter Hansen,et al.  Robust control and model misspecification , 2006, J. Econ. Theory.

[15]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[16]  Kyoungwon Seo,et al.  AMBIGUITY AND SECOND-ORDER BELIEF , 2009 .

[17]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework , 1989 .

[18]  Costis Skiadas.  Scale-invariant asset pricing and consumption/portfolio choice with general attitudes toward risk and uncertainty , 2013 .

[19]  Tomasz Strzalecki,et al.  Axiomatic Foundations of Multiplier Preferences , 2009 .

[20]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[21]  J. Lefoll,et al.  Smooth Ambiguity Aversion and the Continuous-Time Limit , 2011 .

[22]  Martin Schneider,et al.  Recursive multiple-priors , 2003, J. Econ. Theory.

[23]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[24]  E. Fama,et al.  The Equity Premium , 2001 .

[25]  Robert Nau,et al.  Uncertainty aversion with second-order probabilities and utilities , 2001, ISIPTA.

[26]  Ricardo J. Caballero,et al.  Collective Risk Management in a Flight to Quality Episode , 2007 .

[27]  Mark D. Schroder,et al.  Optimal Consumption and Portfolio Selection with Stochastic Differential Utility , 1999 .

[28]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[29]  F. Knight.  The economic nature of the firm: From Risk, Uncertainty, and Profit , 2009 .

[30]  Richard S. Varga,et al.  Proof of Theorem 5 , 1983 .

[31]  Costis Skiadas,et al.  Dynamic Portfolio Choice and Risk Aversion , 2007 .

[32]  Emre Ozdenoren,et al.  Subjective recursive expected utility , 2004 .

[33]  Costis Skiadas.  Robust control and recursive utility , 2003, Finance Stochastics.

[34]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[35]  Darrel,et al.  PDE solutions of stochastic differential utility * , 2001 .

[36]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: An Empirical Analysis , 1991, Journal of Political Economy.

[37]  Evan L. Porteus,et al.  Temporal Resolution of Uncertainty and Dynamic Choice Theory , 1978 .

[38]  Mark Schroder,et al.  OPTIMALITY AND STATE PRICING IN CONSTRAINED FINANCIAL MARKETS WITH RECURSIVE UTILITY UNDER CONTINUOUS AND DISCONTINUOUS INFORMATION , 2008 .

[39]  E. Prescott,et al.  The Equity Risk Premium: A Solution? , 1988 .

[40]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[41]  Larry G. Epstein,et al.  A Two-Person Dynamic Equilibrium under Ambiguity , 2003 .

[42]  Tim Bedford,et al.  The theory of decision making under uncertainty , 2001 .

[43]  P. Weil.  The equity premium puzzle and the risk-free rate puzzle , 1989 .

[44]  M. Royer.  Backward stochastic differential equations with jumps and related non-linear expectations , 2006 .

[45]  Larry G. Epstein,et al.  Ambiguity, risk, and asset returns in continuous time , 2000 .

[46]  T. Sargent,et al.  Robust Control and Model Uncertainty , 2001 .

[47]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[48]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[49]  Chris Shannon,et al.  Sharing risk and ambiguity , 2012, J. Econ. Theory.

[50]  Larry Selden,et al.  A New Representation of Preferences over "Certain A Uncertain" Consumption Pairs: The "Ordinal Certainty Equivalent" Hypothesis , 1978 .

[51]  Haluk Ergin,et al.  A theory of subjective compound lotteries , 2009, J. Econ. Theory.

[52]  P. Brémaud.  Point processes and queues, martingale dynamics , 1983 .

[53]  Robert F. Nau,et al.  Uncertainty Aversion with Second-Order Utilities and Probabilities , 2006, Manag. Sci..

[54]  K. Arrow,et al.  Aspects of the theory of risk-bearing , 1966 .

[55]  Takashi Hayashi,et al.  Intertemporal substitution, risk aversion and ambiguity aversion , 2005 .

[56]  J. Tallon,et al.  Ambiguity and the historical equity premium1 , 2016 .

[57]  P. Protter.  Stochastic integration and differential equations , 1990 .

[58]  Nancy L. Stokey.  The Economics of Inaction: Stochastic Control Models with Fixed Costs , 2008 .

[59]  Larry G. Epstein,et al.  The Structure of Preferences and Attitudes towards the Timing of the Resolution of Uncertainty , 1989 .

[60]  Holger Kraft,et al.  Stochastic Differential Utility as the Continuous-Time Limit of Recursive Utility , 2013, J. Econ. Theory.

[61]  Nate Silver,et al.  The signal and the noise : why so many predictions fail but some don't , 2012 .

[62]  R. Mehra,et al.  THE EQUITY PREMIUM A Puzzle , 1985 .

[63]  Thomas A. Rietz.  The equity risk premium a solution , 1988 .

[64]  Costis Skiadas,et al.  Recursive utility and preferences for information , 1998 .

[65]  K. Arrow.  Essays in the theory of risk-bearing , 1958 .

[66]  E. Prescott,et al.  Investment Under Uncertainty , 1971 .

[67]  J. Keynes.  The General Theory of Employment , 1937 .

[68]  D. Ellsberg.  Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[69]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[70]  T. Bewley.  Knightian decision theory. Part I , 2002 .

[71]  G. Barles,et al.  Backward stochastic differential equations and integral-partial differential equations , 1997 .

[72]  A. Rustichini,et al.  Ambiguity Aversion, Robustness, and the Variational Representation of Preferences , 2006 .

[73]  J. Pratt.  RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .

[74]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[75]  Lars Peter Hansen,et al.  Robustness and ambiguity in continuous time , 2011, J. Econ. Theory.

[76]  Mark D. Schroder,et al.  Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursiv , 2003 .

[77]  M. Marinacci,et al.  A Smooth Model of Decision Making Under Ambiguity , 2003 .

[78]  R. C. Merton,et al.  Continuous-Time Finance , 1990 .

[79]  Tan Wang,et al.  Conditional preferences and updating , 2003, J. Econ. Theory.

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

[81]  Massimo Marinacci,et al.  Alpha As Ambiguity: Robust Mean-Variance Portfolio Analysis , 2010 .

[82]  Jianjun Miao,et al.  Ambiguity, Learning, and Asset Returns , 2009 .

[83]  Jianfeng Zhang.  A numerical scheme for BSDEs , 2004 .

[84]  B. Bouchard,et al.  Discrete time approximation of decoupled Forward-Backward SDE with jumps , 2008 .

[85]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[86]  Massimo Marinacci,et al.  Recursive smooth ambiguity preferences , 2009, J. Econ. Theory.