Contract Theory in Continuous- Time Models

Transcription

1 Jaksa Cvitanic Jianfeng Zhang Contract Theory in Continuous- Time Models fyj Springer

2 Table of Contents Part I Introduction 1 Principal-Agent Problem Problem Formulation Further Reading 6 References 6 2 Single-Period Examples Risk Sharing 7, 2.2 Hidden Action Hidden Type 10 2:4 Further Reading 14 References 14 Part II First Best: Risk Sharing Under Full Information 3 Linear Models with Project Selection, and Preview of Results Linear Dynamics and Control of Volatility The Model Risk Sharing, First Best Solution Implementing the First Best Solution Optimal Contract as a Function of Output Examples Further Reading 24 References 24 4 The General Risk Sharing Problem The Model and the PA Problem Necessary Conditions for Optimality FBSDE Formulation , Adjoint Processes Main Result 28

3 x Table of Contents 4.3 Sufficient Conditions for Optimality Optimal Contracts Implementing the First Best Solution On Uniqueness of Optimal Contracts Examples Linear Dynamics Nonlinear Volatility Selection with Exponential Utilities Linear Contracts Dual Problem A More General Model with Consumption and Recursive Utilities Further Reading "..-.-.' 43 References 43 Part III Second Best: Contracting Under Hidden Action The Case of Moral Hazard 5 Mathematical Theory for General Moral Hazard Problems The Model and the PA Problem Lipschitz Case Agent's Problem Principal's Problem Principal's Problem Based on Principal's Target Actions Principal's Problem Based on Principal's Target Actions: Another Formulation Quadratic Case Agent's Problem Principal's Problem Special Cases Participation Constraint at Time Zero Separable Utility and Participation Constraint at Time Zero Infinite Horizon HJB Approach in Markovian Case A More General Model with Consumption and Recursive Utilities Further Reading.< 83 References 84 6 Special Cases and Applications Exponential Utilities and Lump-Sum Payment The Model Necessary Conditions Derived from the General Theory A Direct Approach A Solvable Special Case with Quadratic Cost General Risk Preferences, Quadratic Cost, and Lump-Sum Payment The Model 94

4 Table of Contents / xi Necessary Conditions Derived from the General Theory A Direct Approach Example: Risk-Neutral Principal and Log-Utility Agent Risk-Neutral Principal and Infinite Horizon The Model Necessary Conditions Derived from the General Theory A Direct Approach Interpretation and Discussion Further Economic Conclusions and Extensions Further Reading 112 References An Application to Capital Structure Problems: Optimal Financing of a Company The Model Agent's Problem Principal's Problem Principal's Problem Under Participation Constraint Properties of the Principal's Value Function Optimal Contract Implementation Using Standard Securities Comparative Statics Example: Agent Owns the Firm Computing Parameter Sensitivities Some Comparative Statics Further Reading 134 References 134 Part IV Third Best: Contracting Under Hidden Action and Hidden Type The Case of Moral Hazard and Adverse Selection 8 Adverse Selection The Model and the PA Problem Constraints Faced by the Principal Quadratic Cost and Lump-Sum Payment Technical Assumptions Solution to the Agent's Problem Principal's Relaxed Problem Properties of the Candidate Optimal Contract Risk-Neutral Agent and Principal Controlling Volatility The Model Main Result: Solving the Relaxed Problem Comparison with the First Best Further Reading 153 References 153

5 Part V Backward SDEs and Forward-Backward SDEs Table of Contents 9 Backward SDEs Introduction Example: Option Pricing and Hedging Linear Backward SDEs Well-PosednessofBSDEs Comparison Theorem and Stability Properties of BSDEs Markovian BSDEs and PDEs Numerical Methods BSDEs with Quadratic Growth Further Reading References Stochastic Maximum Principle Stochastic Control of BSDEs Stochastic Control of FBSDEs Stochastic Control of High-Dimensional BSDEs Stochastic Optimization in Weak Formulation Weak Formulation Versus Strong Formulation Sufficient Conditions in Weak Formulation Necessary Conditions in Weak Formulation Stochastic Optimization for High-Dimensional BSDEs Stochastic Optimization for FBSDEs Some Technical Proofs Heuristic Derivation of the Results of Sect Heuristic Derivation of the Results of Sect Sketch of Proof for Theorem Further Reading 226 References Forward-Backward SDEs FBSDE Definition. ' Fixed Point Approach Four-Step Scheme The Decoupling Approach Method of Continuation Further Reading.\ 247 References 248 References 249 Index 253