© 2013

Contract Theory in Continuous-Time Models


  • Reviewed by international experts

  • Surveys recent results in a systematic way

  • Enables derivation of many qualitative economic conclusions


Part of the Springer Finance book series (FINANCE)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Introduction

    1. Front Matter
      Pages 1-1
    2. Jakša Cvitanić, Jianfeng Zhang
      Pages 3-6
    3. Jakša Cvitanić, Jianfeng Zhang
      Pages 7-14
  3. First Best: Risk Sharing Under Full Information

    1. Front Matter
      Pages 15-15
    2. Jakša Cvitanić, Jianfeng Zhang
      Pages 17-24
    3. Jakša Cvitanić, Jianfeng Zhang
      Pages 25-43
  4. Second Best: Contracting Under Hidden Action—The Case of Moral Hazard

    1. Front Matter
      Pages 45-45
    2. Jakša Cvitanić, Jianfeng Zhang
      Pages 47-84
    3. Jakša Cvitanić, Jianfeng Zhang
      Pages 85-113
  5. Third Best: Contracting Under Hidden Action and Hidden Type—The Case of Moral Hazard and Adverse Selection

    1. Front Matter
      Pages 135-135
    2. Jakša Cvitanić, Jianfeng Zhang
      Pages 137-153
  6. Backward SDEs and Forward-Backward SDEs

    1. Front Matter
      Pages 155-155
    2. Jakša Cvitanić, Jianfeng Zhang
      Pages 157-182
    3. Jakša Cvitanić, Jianfeng Zhang
      Pages 183-227
    4. Jakša Cvitanić, Jianfeng Zhang
      Pages 229-248
  7. Back Matter
    Pages 249-255

About this book


In recent years there has been a significant increase of interest in continuous-time Principal-Agent models, or contract theory, and their applications. Continuous-time models provide a powerful and elegant framework for solving stochastic optimization problems of finding the optimal contracts between two parties, under various assumptions on the information they have access to, and the effect they have on the underlying "profit/loss" values. This monograph surveys recent results of the theory in a systematic way, using the approach of the so-called Stochastic Maximum Principle, in models driven by Brownian Motion.

Optimal contracts are characterized via a system of Forward-Backward Stochastic Differential Equations. In a number of interesting special cases these can be solved explicitly, enabling derivation of many qualitative economic conclusions.


91G80, 93E20 forward-backward SDEs optimal contracts principal-agent problems quantitative finance stochastic maximum principle

Authors and affiliations

  1. 1.Div. of the Humanities, Social ScienceCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

About the authors

Jakša Cvitanić held positions at Columbia University (Statistics), University of Southern California (Mathematics and Economics), and currently at Caltech (Social Sciences). He has served on the editorial boards of journals in the areas of Financial Mathematics, Applied Probability and Optimization, as well as on the Council of the Bachelier Finance Society. Jianfeng Zhang is currently associate professor at the University of Southern California (Mathematics Department).

Bibliographic information

Industry Sectors
Finance, Business & Banking


“The book under review provides a complete treatment of the principal-agent problem that covers all cases treated in economic literature … . It is the first of its kind in that it provides a fully developed mathematical framework addressing the principal-agent problem with complete proofs and explanations of all mathematical tools used therein. … The introduction of this book is accessible to a general audience.” (Olympia Hadjiliadis, Bulletin of the American Mathematical Society, Vol. 52 (3), July, 2015)

“The present book presents a nice exposition of the theory of the stochastic maximum principle, starting with BSDEs, and of its applications to contract theory. … I recommend it to anyone working on or teaching the mathematical aspects of contract theory and/or stochastic control.” (Etienne Pardoux, SIAM Review, Vol. 57 (2), June, 2015)

“This book considers contract theory in continuous time. … This book is a good reference book for researchers and graduate students in economic theory, finance and mathematical economics. Continuous-time contract theory is particularly useful in finance. This book provides a basic methodological framework, which can be used to develop further advances, both in applications and in theory.” (Susheng Wang, Mathematical Reviews, August, 2013)