Relaxed Utility Maximization in Complete Markets
|
|
- Theodora Hood
- 5 years ago
- Views:
Transcription
1 Relaxed Utility Maximization in Complete Markets Paolo Guasoni (Joint work with Sara Biagini) Boston University and Dublin City University Analysis, Stochastics, and Applications In Honor of Walter Schachermayer July 15 th, 2010
2 Outline Relaxing what? Preferences: risk aversion vanishing as wealth increases. Payoffs: more than random variables. Problem: Utility maximization in a complete market. Asymptotic elasticity of utility function can approach one. Solution: Add topology to probability space. Payoffs as measures. Classic payoffs as densities. Results: Expected utility representation. Singular utility. Characterization of optimal solutions.
3 The Usual Argument Utility Maximization from terminal wealth: max{e P [U(X)] : E Q [X] x} Use first-order condition to look for solution: U ( ˆX) = y dq dp Pick the Lagrange multiplier y which saturates constraint: If there is any. Assumptions on U? E Q [ ˆX(y) ] = x
4 The Usual Conditions Karatzas, Lehoczky, Shreve, and Xu (1991): U (βx) < αu (x) for all x > x 0 > 0 and some α < 1 < β This condition implies the next one. Kramkov and Schachemayer (1999): AE(U) = lim sup x xu (x) U(x) < 1 Guarantees an optimal payoff in any market model. Condition not satisfied? No solution for some model. Interpretation?
5 Asymptotic Relative Risk Aversion What do these conditions mean (and imply)? Suppose Relative Risk Aversion has a limit: ARRA(U) = lim x xu (x) U (x) Then AE(U) < 1 is equivalent to ARRA(U) > 0. As wealth increases, risk aversion must remain above ε > 0. Why? Lower risk premium when you are rich? AE(U) = 1 as Asymptotic Relative Risk Neutrality. Relative Risk Aversion positive. But declines to zero. Relaxed Investor. Relevance?
6 Who Cares? Logarithmic, Power, and Exponential utilities satisfy ARRA(U) > 0. Why bother about ARRA(U) = 0, if there are no examples? Heterogeneous preferences equilibria. Benninga and Mayshar (2000), Cvitanic and Malamud (2008). Complete market with several power utility agents. Power of utility depends on agent. Utility function of representative agent. Relative risk aversion decreases to that of least risk averse agent. All values of relative risk aversion present in the market? Risk aversion of representative agent decreases to zero. Asymptotic elasticty equals one. Solution may not exist. But why?
7 Singular Investment Kramkov and Schachermayer (1999) show what goes wrong. Countable space Ω = (ω n ) n 1. dp/dq(ω n ) = p n /q n as n. Finite space Ω N. ω N n = ω n for n < N. (ω n ) n N lumped into ω N N. Solution exists in each Ω N. Satisfies first order condition: U (X N n ) = yq n /p n 1 n < N U (X N N ) = yq N/p N where p N N = 1 N 1 n=1 p n and q N N = 1 N 1 n=1 q n. What happens to (X N n ) 1 n N as N? X N n X n, which solves U (X n ) = yq n /p n for n 1. For large initial wealth x, E Q [X] < x. Where has x E Q [X] gone? qn NX N N converges to x E Q [X]. But qn N decreases to 0. Invest x E Q [X] in a payoff equal to with 0 probability.
8 Main Idea The problem wants to concentrate money on null sets. But expected utility does not see such sets. Relax the notion of payoff. Relax utility functional. Do it consistently.
9 Setting (Ω, T ) Polish space. P, Q Borel-regular probabilities on Borel σ-field F. Q P Payoffs available with initial capital x: C(x) := {X L 0 + E Q[X] x} Market complete. U : (0, + ) (, + ) strictly increasing, strictly concave, continuously differentiable. Inada conditions U (0 + ) = + and U (+ ) = 0. sup X C(x) E P [U(X)] < U( ) P (and hence Q) has full support, i.e. P(G) > 0 for any open set G. If not, replace Ω with support of P.
10 Relaxed Payoffs Definition A relaxed payoff is an element of D(x), the weak star σ(rba(ω), C b (Ω)) closed set {µ rba(ω) + µ(ω) x}. rba(ω): Borel regular, finitely additive signed measures on Ω. Isometric to (C b (Ω)). µ rba(ω) admits unique decomposition: µ = µ a + µ s + µ p, µ a Q and µ s Q countably additive. µ p purely finitely additive. All components Borel regular.
11 Finitely Additive? Dubious interpretation of finitely additive measures as payoffs. Allow them a priori. For technical convenience. Let the problem rule them out. They are not optimal anyway.
12 Relaxed Utility Relaxed utility map I U : rba(ω) [, + ). Defined on rba(ω) as upper semicontinuous envelope of I U : I U (µ) = inf{g(µ) G weak u.s.c., G I U on L 1 (Q)}. Relaxed utility maximization problem: max µ D(x) I U(µ) Relaxed utility map I U weak star upper semicontinuous. Space of relaxed payoffs D(x) weak star compact. Relaxed utility maximization has solution by construction. Elaborate tautology. Find concrete formula for I U. Integral representation.
13 Singular Utility V (y) = sup x>0 (U(x) xy) convex conjugate of U. Singular utility: nonnegative function ϕ defined as: { ( ϕ(ω) = inf g(ω) g C b(ω), E P [V g dq )] dp } <, Upper semi-continuous, as infimum of continuous functions. Defined for all ω. Function, not random variable. W : Ω R + R sup-convolution of U and x xϕ(ω) dq ( W (ω, x) := sup U(z) + (x z)ϕ(ω) dq z x dp (ω) dp (ω): ). ϕ(ω) = 0 for each ω where dp/dq is bounded in a neighborhood. Concentrating wealth suboptimal if odds finite. ϕ may be positive only on poles of dp/dq.
14 Theorem Integral Representation Let µ rba(ω) +, and Q P fully supported probabilities. i) In general: I U (µ) = E P [W (, dµ )] a + dq ϕdµ s + inf f C b (Ω),E P[V(f dq dp )]< µ p (f ). ii) If ϕ = 0 P-a.s., then: I U (µ) = E P [U ( dµa dq )] + ϕdµ s + inf f C b (Ω),E P[V(f dq dp )]< µ p (f ). iii) If lim sup x xu (x) U(x) < 1, then {ϕ = 0} = Ω and I U (µ) = E P [U ( )] dµa. dq
15 Three Parts First formula holds for any µ rba(ω) +. But has finitely additive part......and has sup-convolution W instead of U. Second formula replaces W with U under additional assumption. Then utility is sum of three pieces. Usual expected utility E[U(X)] with X = dµa dq. Finitely additive part. Singular utility ϕdµ s. Accounts for utility from concentration of wealth on P-null sets. ϕ(ω) represents maximal utility from Dirac delta on ω Only usual utility remains for AE(U) < 1.
16 Proof Strategy Separate countably additive from purely finitely additive part: I U (µ) = I U (µ c ) + inf µ p(f ). f Dom(J V ) Find integral representation for countably additive part. Separate absolutely continuous and singular components. Identify absolutely continuous part as original expected utility map, and singular part as asymptotic utility.
17 Coercivity Assumption Set y 0 = sup ω Ω ϕ(ω). Assume that either y 0 = 0, or there exist ε > 0 and g ( C b (Ω))] such that the closed set K = {g y 0 ε} is compact and E P [V <. g dq dp Maximizing sequences for singular utility do not escape compacts. Automatic if Ω compact. In general, first find ϕ......and check its maximizing sequences. Standard coercitivy condition. Counterexamples without it.
18 Relaxed utility Maximization Theorem Under coercivity assumption, and if ϕ = 0 a.s.: i) u(x) = max µ D(x) I U (µ); ii) u(x) = E[U(X (x))] + ϕdµ s, where X (x) = dµ a dq. iii) Budget constraint binding: µ (Ω) = E Q [X (x)] + µ s(ω) = x. iv) µ a unique. Support of any µ s satisfies: supp(µ s) argmax(ϕ). v) If x > x 0, any solution has the form µ = µ a + µ s, where µ s(ω) = x x 0. vi) u(x) = u(x 0 ) + (x x 0 ) max ω ϕ(ω) = u(x 0 ) + (x x 0 )y 0.
19 Conclusion Happy Birthday for your first 60! Ad Maiora et Meliora!
RELAXED UTILITY MAXIMIZATION IN COMPLETE MARKETS. SARA BIAGINI University of Pisa. PAOLO GUASONI Boston University
RELAXED UTILITY MAXIMIZATION IN COMPLETE MARKETS SARA BIAGINI University of Pisa PAOLO GUASONI Boston University For a relaxed investor one whose relative risk aversion vanishes as wealth becomes large
More informationRelaxed Utility Maximization
Relaxed Utility Maximization Sara Biagini Paolo Guasoni July 15, 2008 Abstract For utility functions U : R + R Kramkov and Schachermayer [KS99] showed that under a condition on U, the well- known condition
More informationOn the dual problem of utility maximization
On the dual problem of utility maximization Yiqing LIN Joint work with L. GU and J. YANG University of Vienna Sept. 2nd 2015 Workshop Advanced methods in financial mathematics Angers 1 Introduction Basic
More informationSensitivity analysis of the expected utility maximization problem with respect to model perturbations
Sensitivity analysis of the expected utility maximization problem with respect to model perturbations Mihai Sîrbu, The University of Texas at Austin based on joint work with Oleksii Mostovyi University
More informationNecessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs
Necessary Conditions for the Existence of Utility Maximizing Strategies under Transaction Costs Paolo Guasoni Boston University and University of Pisa Walter Schachermayer Vienna University of Technology
More informationMinimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization
Finance and Stochastics manuscript No. (will be inserted by the editor) Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization Nicholas Westray Harry Zheng. Received: date
More informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
More informationTHE ASYMPTOTIC ELASTICITY OF UTILITY FUNCTIONS AND OPTIMAL INVESTMENT IN INCOMPLETE MARKETS 1
The Annals of Applied Probability 1999, Vol. 9, No. 3, 94 95 THE ASYMPTOTIC ELASTICITY OF UTILITY FUNCTIONS AND OPTIMAL INVESTMENT IN INCOMPLETE MARKETS 1 By D. Kramkov 2 and W. Schachermayer Steklov Mathematical
More informationConjugate duality in stochastic optimization
Ari-Pekka Perkkiö, Institute of Mathematics, Aalto University Ph.D. instructor/joint work with Teemu Pennanen, Institute of Mathematics, Aalto University March 15th 2010 1 / 13 We study convex problems.
More information02. Measure and integral. 1. Borel-measurable functions and pointwise limits
(October 3, 2017) 02. Measure and integral Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2017-18/02 measure and integral.pdf]
More informationLecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016
Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Entropy Since this course is about entropy maximization,
More informationEfficient portfolios in financial markets with proportional transaction costs
Joint work E. Jouini and V. Portes Conference in honour of Walter Schachermayer, July 2010 Contents 1 2 3 4 : An efficient portfolio is an admissible portfolio which is optimal for at least one agent.
More informationLectures for the Course on Foundations of Mathematical Finance
Definitions and properties of Lectures for the Course on Foundations of Mathematical Finance First Part: Convex Marco Frittelli Milano University The Fields Institute, Toronto, April 2010 Definitions and
More informationSuper-replication and utility maximization in large financial markets
Super-replication and utility maximization in large financial markets M. De Donno P. Guasoni, M. Pratelli, Abstract We study the problems of super-replication and utility maximization from terminal wealth
More informationMarket environments, stability and equlibria
Market environments, stability and equlibria Gordan Žitković Department of Mathematics University of exas at Austin Austin, Aug 03, 2009 - Summer School in Mathematical Finance he Information Flow two
More informationUtility Maximization in Incomplete Markets with Random Endowment. Walter Schachermayer Hui Wang
Utility Maximization in Incomplete Markets with Random Endowment Jakša Cvitanić Walter Schachermayer Hui Wang Working Paper No. 64 Januar 2000 Januar 2000 SFB Adaptive Information Systems and Modelling
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationTHE ARROW PRATT INDEXES OF RISK AVERSION AND CONVEX RISK MEASURES THEY IMPLY
THE ARROW PRATT INDEXES OF RISK AVERSION AND CONVEX RISK MEASURES THEY IMPLY PAUL C. KETTLER ABSTRACT. The Arrow Pratt index of relative risk aversion combines the important economic concepts of elasticity
More informationEconomics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
More informationWHY SATURATED PROBABILITY SPACES ARE NECESSARY
WHY SATURATED PROBABILITY SPACES ARE NECESSARY H. JEROME KEISLER AND YENENG SUN Abstract. An atomless probability space (Ω, A, P ) is said to have the saturation property for a probability measure µ on
More informationPortfolio Optimization in discrete time
Portfolio Optimization in discrete time Wolfgang J. Runggaldier Dipartimento di Matematica Pura ed Applicata Universitá di Padova, Padova http://www.math.unipd.it/runggaldier/index.html Abstract he paper
More informationHealthcare and Consumption with Aging
Healthcare and Consumption with Aging Yu-Jui Huang University of Colorado Joint work with Paolo Guasoni Boston University and Dublin City University University of Colorado September 9, 2016 This is research
More informationHicksian Demand and Expenditure Function Duality, Slutsky Equation
Hicksian Demand and Expenditure Function Duality, Slutsky Equation Econ 2100 Fall 2017 Lecture 6, September 14 Outline 1 Applications of Envelope Theorem 2 Hicksian Demand 3 Duality 4 Connections between
More informationPORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS
International Journal of Theoretical and Applied Finance c World Scientific Publishing Company PORTFOLIO OPTIMIZATION WITH PERFORMANCE RATIOS HONGCAN LIN Department of Statistics and Actuarial Science,
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationMarch 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth
More informationHomework Assignment #2 for Prob-Stats, Fall 2018 Due date: Monday, October 22, 2018
Homework Assignment #2 for Prob-Stats, Fall 2018 Due date: Monday, October 22, 2018 Topics: consistent estimators; sub-σ-fields and partial observations; Doob s theorem about sub-σ-field measurability;
More informationIntroduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence
Introduction to Empirical Processes and Semiparametric Inference Lecture 08: Stochastic Convergence Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and Operations
More informationGeometry and Optimization of Relative Arbitrage
Geometry and Optimization of Relative Arbitrage Ting-Kam Leonard Wong joint work with Soumik Pal Department of Mathematics, University of Washington Financial/Actuarial Mathematic Seminar, University of
More informationDuality and Utility Maximization
Duality and Utility Maximization Bachelor Thesis Niklas A. Pfister July 11, 2013 Advisor: Prof. Dr. Halil Mete Soner Department of Mathematics, ETH Zürich Abstract This thesis explores the problem of maximizing
More information5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.
5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint
More information6.254 : Game Theory with Engineering Applications Lecture 7: Supermodular Games
6.254 : Game Theory with Engineering Applications Lecture 7: Asu Ozdaglar MIT February 25, 2010 1 Introduction Outline Uniqueness of a Pure Nash Equilibrium for Continuous Games Reading: Rosen J.B., Existence
More informationFINANCIAL OPTIMIZATION
FINANCIAL OPTIMIZATION Lecture 1: General Principles and Analytic Optimization Philip H. Dybvig Washington University Saint Louis, Missouri Copyright c Philip H. Dybvig 2008 Choose x R N to minimize f(x)
More informationThomas Knispel Leibniz Universität Hannover
Optimal long term investment under model ambiguity Optimal long term investment under model ambiguity homas Knispel Leibniz Universität Hannover knispel@stochastik.uni-hannover.de AnStAp0 Vienna, July
More informationMultivariate Utility Maximization with Proportional Transaction Costs
Multivariate Utility Maximization with Proportional Transaction Costs Luciano Campi, Mark Owen To cite this version: Luciano Campi, Mark Owen. Multivariate Utility Maximization with Proportional Transaction
More informationRisk-Averse Dynamic Optimization. Andrzej Ruszczyński. Research supported by the NSF award CMMI
Research supported by the NSF award CMMI-0965689 Outline 1 Risk-Averse Preferences 2 Coherent Risk Measures 3 Dynamic Risk Measurement 4 Markov Risk Measures 5 Risk-Averse Control Problems 6 Solution Methods
More informationis a Borel subset of S Θ for each c R (Bertsekas and Shreve, 1978, Proposition 7.36) This always holds in practical applications.
Stat 811 Lecture Notes The Wald Consistency Theorem Charles J. Geyer April 9, 01 1 Analyticity Assumptions Let { f θ : θ Θ } be a family of subprobability densities 1 with respect to a measure µ on a measurable
More informationREGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS
REGULAR LAGRANGE MULTIPLIERS FOR CONTROL PROBLEMS WITH MIXED POINTWISE CONTROL-STATE CONSTRAINTS fredi tröltzsch 1 Abstract. A class of quadratic optimization problems in Hilbert spaces is considered,
More informationIntroduction to Game Theory: Simple Decisions Models
Introduction to Game Theory: Simple Decisions Models John C.S. Lui Department of Computer Science & Engineering The Chinese University of Hong Kong John C.S. Lui (CUHK) Advanced Topics in Network Analysis
More informationOn duality theory of conic linear problems
On duality theory of conic linear problems Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 3332-25, USA e-mail: ashapiro@isye.gatech.edu
More information1: PROBABILITY REVIEW
1: PROBABILITY REVIEW Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) Slides 1: Probability Review 1 / 56 Outline We will review the following
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationLecture 4 Lebesgue spaces and inequalities
Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how
More informationSEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE
SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE FRANÇOIS BOLLEY Abstract. In this note we prove in an elementary way that the Wasserstein distances, which play a basic role in optimal transportation
More informationRandom G -Expectations
Random G -Expectations Marcel Nutz ETH Zurich New advances in Backward SDEs for nancial engineering applications Tamerza, Tunisia, 28.10.2010 Marcel Nutz (ETH) Random G-Expectations 1 / 17 Outline 1 Random
More informationAN INTRODUCTION TO MATHEMATICAL ANALYSIS ECONOMIC THEORY AND ECONOMETRICS
AN INTRODUCTION TO MATHEMATICAL ANALYSIS FOR ECONOMIC THEORY AND ECONOMETRICS Dean Corbae Maxwell B. Stinchcombe Juraj Zeman PRINCETON UNIVERSITY PRESS Princeton and Oxford Contents Preface User's Guide
More informationA BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras.
A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM STEVO TODORCEVIC Abstract. We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition. MSC 2000:
More informationarxiv:submit/ [q-fin.pm] 25 Sep 2011
arxiv:submit/0324648 [q-fin.pm] 25 Sep 2011 Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung Qingshuo Song Jie Yang September 25, 2011 Abstract We study
More information2 (Bonus). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due 9/5). Prove that every countable set A is measurable and µ(a) = 0. 2 (Bonus). Let A consist of points (x, y) such that either x or y is
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationStability of optimization problems with stochastic dominance constraints
Stability of optimization problems with stochastic dominance constraints D. Dentcheva and W. Römisch Stevens Institute of Technology, Hoboken Humboldt-University Berlin www.math.hu-berlin.de/~romisch SIAM
More informationRisk Minimization under Transaction Costs
Noname manuscript No. (will be inserted by the editor) Risk Minimization under Transaction Costs Paolo Guasoni Bank of Italy Research Department Via Nazionale, 91 00184 Roma e-mail: guasoni@dm.unipi.it
More informationRobust preferences and robust portfolio choice
Robust preferences and robust portfolio choice Hans FÖLLMER Institut für Mathematik Humboldt-Universität Unter den Linden 6 10099 Berlin, Germany foellmer@math.hu-berlin.de Alexander SCHIED School of ORIE
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationWeak convergence. Amsterdam, 13 November Leiden University. Limit theorems. Shota Gugushvili. Generalities. Criteria
Weak Leiden University Amsterdam, 13 November 2013 Outline 1 2 3 4 5 6 7 Definition Definition Let µ, µ 1, µ 2,... be probability measures on (R, B). It is said that µ n converges weakly to µ, and we then
More informationFunctional Properties of MMSE
Functional Properties of MMSE Yihong Wu epartment of Electrical Engineering Princeton University Princeton, NJ 08544, USA Email: yihongwu@princeton.edu Sergio Verdú epartment of Electrical Engineering
More informationA monetary value for initial information in portfolio optimization
A monetary value for initial information in portfolio optimization this version: March 15, 2002; Jürgen Amendinger 1, Dirk Becherer 2, Martin Schweizer 3 1 HypoVereinsbank AG, International Markets, Equity
More informationChoice under Uncertainty
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) Group 2 Dr. S. Farshad Fatemi Chapter 6: Choice under Uncertainty
More informationProperties of Walrasian Demand
Properties of Walrasian Demand Econ 2100 Fall 2017 Lecture 5, September 12 Problem Set 2 is due in Kelly s mailbox by 5pm today Outline 1 Properties of Walrasian Demand 2 Indirect Utility Function 3 Envelope
More informationOPTIMAL INVESTMENT WITH RANDOM ENDOWMENTS IN INCOMPLETE MARKETS
The Annals of Applied Probability 2004, Vol. 14, No. 2, 845 864 Institute of Mathematical Statistics, 2004 OPTIMAL INVESTMENT WITH RANDOM ENDOWMENTS IN INCOMPLETE MARKETS BY JULIEN HUGONNIER 1 AND DMITRY
More informationMATH 205C: STATIONARY PHASE LEMMA
MATH 205C: STATIONARY PHASE LEMMA For ω, consider an integral of the form I(ω) = e iωf(x) u(x) dx, where u Cc (R n ) complex valued, with support in a compact set K, and f C (R n ) real valued. Thus, I(ω)
More informationOn fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems
On fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems Jose Cánovas, Jiří Kupka* *) Institute for Research and Applications of Fuzzy Modeling University of Ostrava Ostrava, Czech
More informationEconomics 401 Sample questions 2
Economics 401 Sample questions 1. What does it mean to say that preferences fit the Gorman polar form? Do quasilinear preferences fit the Gorman form? Do aggregate demands based on the Gorman form have
More informationCould Nash equilibria exist if the payoff functions are not quasi-concave?
Could Nash equilibria exist if the payoff functions are not quasi-concave? (Very preliminary version) Bich philippe Abstract In a recent but well known paper (see [11]), Reny has proved the existence of
More informationarxiv: v1 [q-fin.pr] 4 Jun 2007
Stability of utility-maximization in incomplete markets arxiv:76.474v1 [q-fin.pr] 4 Jun 27 Kasper Larsen Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, kasperl@andrew.cmu.edu
More informationReview of Optimization Methods
Review of Optimization Methods Prof. Manuela Pedio 20550 Quantitative Methods for Finance August 2018 Outline of the Course Lectures 1 and 2 (3 hours, in class): Linear and non-linear functions on Limits,
More informationConvergence of utility indifference prices to the superreplication price in a multiple-priors framework
Convergence of utility indifference prices to the superreplication price in a multiple-priors framework Romain Blanchard, Laurence Carassus o cite this version: Romain Blanchard, Laurence Carassus. Convergence
More informationarxiv: v1 [math.pr] 16 Jun 2008
The Annals of Applied Probability 2008, Vol. 18, No. 3, 929 966 DOI: 10.1214/07-AAP469 c Institute of Mathematical Statistics, 2008 arxiv:0806.2582v1 [math.pr] 16 Jun 2008 A UNIFIED FRAMEWORK FOR UTILITY
More informationWe suppose that for each "small market" there exists a probability measure Q n on F n that is equivalent to the original measure P n, suchthats n is a
Asymptotic Arbitrage in Non-Complete Large Financial Markets Irene Klein Walter Schachermayer Institut fur Statistik, Universitat Wien Abstract. Kabanov and Kramkov introduced the notion of "large nancial
More informationIncreases in Risk Aversion and the Distribution of Portfolio Payoffs
Increases in Risk Aversion and the Distribution of Portfolio Payoffs Philip H. Dybvig Yajun Wang July 14, 2010 Abstract In this paper, we derive new comparative statics results in the distribution of portfolio
More informationArrow-Debreu Equilibria for Rank-Dependent Utilities with Heterogeneous Probability Weighting
Arrow-Debreu Equilibria for Rank-Dependent Utilities with Heterogeneous Probability Weighting Hanqing Jin Jianming Xia Xun Yu Zhou August 12, 2016 Abstract We study Arrow Debreu equilibria for a one-period-two-date
More informationA new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009
A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance
More informationSimultaneous Gaussian quadrature for Angelesco systems
for Angelesco systems 1 KU Leuven, Belgium SANUM March 22, 2016 1 Joint work with Doron Lubinsky Introduced by C.F. Borges in 1994 Introduced by C.F. Borges in 1994 (goes back to Angelesco 1918). Introduced
More informationOptimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112
Optimal Control Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Review of the Theory of Optimal Control Section 1 Review of the Theory of Optimal Control Ömer
More informationDuality in constrained optimal investment and consumption problems: a synopsis 1
Duality in constrained optimal investment and consumption problems: a synopsis 1 I. Klein 2 and L.C.G. Rogers 3 Abstract In the style of Rogers (21), we give a unified method for finding the dual problem
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationCVaR and Examples of Deviation Risk Measures
CVaR and Examples of Deviation Risk Measures Jakub Černý Department of Probability and Mathematical Statistics Stochastic Modelling in Economics and Finance November 10, 2014 1 / 25 Contents CVaR - Dual
More informationPATH FUNCTIONALS OVER WASSERSTEIN SPACES. Giuseppe Buttazzo. Dipartimento di Matematica Università di Pisa.
PATH FUNCTIONALS OVER WASSERSTEIN SPACES Giuseppe Buttazzo Dipartimento di Matematica Università di Pisa buttazzo@dm.unipi.it http://cvgmt.sns.it ENS Ker-Lann October 21-23, 2004 Several natural structures
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationGeneral Theory of Large Deviations
Chapter 30 General Theory of Large Deviations A family of random variables follows the large deviations principle if the probability of the variables falling into bad sets, representing large deviations
More informationIntegral Jensen inequality
Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationEconomics 201B Economic Theory (Spring 2017) Bargaining. Topics: the axiomatic approach (OR 15) and the strategic approach (OR 7).
Economics 201B Economic Theory (Spring 2017) Bargaining Topics: the axiomatic approach (OR 15) and the strategic approach (OR 7). The axiomatic approach (OR 15) Nash s (1950) work is the starting point
More informationMoment Measures. Bo az Klartag. Tel Aviv University. Talk at the asymptotic geometric analysis seminar. Tel Aviv, May 2013
Tel Aviv University Talk at the asymptotic geometric analysis seminar Tel Aviv, May 2013 Joint work with Dario Cordero-Erausquin. A bijection We present a correspondence between convex functions and Borel
More informationarxiv: v2 [math.pr] 12 May 2013
arxiv:1304.3284v2 [math.pr] 12 May 2013 Existence and uniqueness of Arrow-Debreu equilibria with consumptions in L 0 + Dmitry Kramkov Carnegie Mellon University and University of Oxford, Department of
More informationUTILITY OPTIMIZATION IN A FINITE SCENARIO SETTING
UTILITY OPTIMIZATION IN A FINITE SCENARIO SETTING J. TEICHMANN Abstract. We introduce the main concepts of duality theory for utility optimization in a setting of finitely many economic scenarios. 1. Utility
More informationWorst Case Portfolio Optimization and HJB-Systems
Worst Case Portfolio Optimization and HJB-Systems Ralf Korn and Mogens Steffensen Abstract We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent,
More informationA Note on Robust Representations of Law-Invariant Quasiconvex Functions
A Note on Robust Representations of Law-Invariant Quasiconvex Functions Samuel Drapeau Michael Kupper Ranja Reda October 6, 21 We give robust representations of law-invariant monotone quasiconvex functions.
More informationThe Non-Existence of Representative Agents
The Non-Existence of Representative Agents Matthew O. Jackson and Leeat Yariv November 2015 Abstract We characterize environments in which there exists a representative agent: an agent who inherits the
More informationBandits : optimality in exponential families
Bandits : optimality in exponential families Odalric-Ambrym Maillard IHES, January 2016 Odalric-Ambrym Maillard Bandits 1 / 40 Introduction 1 Stochastic multi-armed bandits 2 Boundary crossing probabilities
More informationOn the shape of solutions to the Extended Fisher-Kolmogorov equation
On the shape of solutions to the Extended Fisher-Kolmogorov equation Alberto Saldaña ( joint work with Denis Bonheure and Juraj Földes ) Karlsruhe, December 1 2015 Introduction Consider the Allen-Cahn
More informationConvex duality in optimal investment and contingent claim valuation in illiquid markets
Convex duality in optimal investment and contingent claim valuation in illiquid markets Teemu Pennanen Ari-Pekka Perkkiö March 9, 2016 Abstract This paper studies convex duality in optimal investment and
More informationLarge Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials
Large Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials Maxim L. Yattselev joint work with Christopher D. Sinclair International Conference on Approximation
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationFinancial Asset Price Bubbles under Model Uncertainty
Financial Asset Price Bubbles under Model Uncertainty Francesca Biagini, Jacopo Mancin October 24, 27 Abstract We study the concept of financial bubbles in a market model endowed with a set P of probability
More informationThis corresponds to a within-subject experiment: see same subject make choices from different menus.
Testing Revealed Preference Theory, I: Methodology The revealed preference theory developed last time applied to a single agent. This corresponds to a within-subject experiment: see same subject make choices
More informationLecture 2: Random Variables and Expectation
Econ 514: Probability and Statistics Lecture 2: Random Variables and Expectation Definition of function: Given sets X and Y, a function f with domain X and image Y is a rule that assigns to every x X one
More informationBayesian Persuasion Online Appendix
Bayesian Persuasion Online Appendix Emir Kamenica and Matthew Gentzkow University of Chicago June 2010 1 Persuasion mechanisms In this paper we study a particular game where Sender chooses a signal π whose
More informationNear-Potential Games: Geometry and Dynamics
Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo January 29, 2012 Abstract Potential games are a special class of games for which many adaptive user dynamics
More information