Time-Inconsistent Control Theory with Finance Applications

This book is devoted to problems of stochastic control and stopping that are time inconsistent in the sense that they do not admit a Bellman optimality principle. These problems are cast in a game-theoretic framework, with the focus on subgame-perfect Nash equilibrium strategies. The general theory is illustrated with a number of finance applications.

In dynamic choice problems, time inconsistency is the rule rather than the exception. Indeed, as Robert H. Strotz pointed out in his seminal 1955 paper, relaxing the widely used ad hoc assumption of exponential discounting gives rise to time inconsistency. Other famous examples of time inconsistency include mean-variance portfolio choice and prospect theory in a dynamic context. For such models, the very concept of optimality becomes problematic, as the decision maker’s preferences change over time in a temporally inconsistent way. In this book, a time-inconsistent problem is viewed as a non-cooperative game between the agent’s currentand future selves, with the objective of finding intrapersonal equilibria in the game-theoretic sense. A range of finance applications are provided, including problems with non-exponential discounting, mean-variance objective, time-inconsistent linear quadratic regulator, probability distortion, and market equilibrium with time-inconsistent preferences.

Time-Inconsistent Control Theory with Finance Applications offers the first comprehensive treatment of time-inconsistent control and stopping problems, in both continuous and discrete time, and in the context of finance applications. Intended for researchers and graduate students in the fields of finance and economics, it includes a review of the standard time-consistent results, bibliographical notes, as well as detailed examples showcasing time inconsistency problems. For the reader unacquainted with standard arbitrage theory, an appendix provides a toolbox of material needed for the book.

1 Introduction.- Part I Optimal Control in Discrete Time.- 2 Dynamic Programming Theory.- 3 The Linear Quadratic Regulator.- 4 A Simple Equilibrium Model.- Part II Time-Inconsistent Control in Discrete Time.- 5 Time-Inconsistent Control Theory.- 6 Extensions and Further Results.- 7 Non-Exponential Discounting.- 8 Mean-Variance Portfolios.- 9 Time-Inconsistent Regulator Problems.- 10 A Time-Inconsistent Equilibrium Model.- Part III Optimal Control in Continuous Time.- 11 Dynamic Programming Theory.- 12 The Continuous-Time Linear Quadratic Regulator.- 13 Optimal Consumption and Investment.- 14 A Simple Equilibrium Model.- Part IV Time-Inconsistent Control in Continuous Time.- 15 Time-Inconsistent Control Theory.- 16 Special Cases and Extensions.- 17 Non-Exponential Discounting.- 18 Mean-Variance Control.- 19 The Inconsistent Linear Quadratic Regulator.- 20 A Time-Inconsistent Equilibrium Model.- Part V Optimal Stopping Theory.- 21 Optimal Stopping in Discrete Time.- 22 Optimal Stopping in Continuous Time.- Part VI Time-Inconsistent Stopping Problems.- 23 Time-Inconsistent Stopping in Discrete Time.- 24 Time-Inconsistent Stopping in Continuous Time.- 25 Time-Inconsistent Stopping Under Distorted Probabilities.- A Basic Arbitrage Theory.- References.

Tomas Björk was a Professor of Mathematical Finance at the Stockholm School of Economics. He was also affiliated with KTH (Royal Institute of Technology), Sweden, and Aarhus University, Denmark. Tomas served as president of the Bachelier Finance Society, co-editor of the journal Mathematical Finance, and a member of the editorial board of Finance and Stochastics and other journals. He published numerous journal articles on mathematical finance, and in particular is known for his research on point process driven forward-rate models, consistent forward-rate curves, general interest-rate theory, finite-dimensional realizations of infinite-dimensional SDEs, good deal bounds, and time-inconsistent control theory. Tomas was the author of the widely used and influential textbook, Arbitrage Theory in Continuous Time, which is now in its fourth edition. He passed away in 2021.

Mariana Khapko is an Assistant Professor of Finance at the University of Toronto,Canada. She is also an affiliated Research Fellow of the Swedish House of Finance at the Stockholm School of Economics. Her research focuses on financial mathematics and financial markets. She has published articles on time-inconsistent control theory, asset pricing and portfolio choice, information in securities markets, and financial market design. Mariana obtained her PhD in Finance from the Stockholm School of Economics, under the supervision of Tomas Björk.

Agatha Murgoci currently works as a senior quantitative developer at Ørsted, a leading off-shore wind energy company. Prior to this, she was an assistant professor at Copenhagen Business School and at Aarhus University, Denmark. She has published papers on time-inconsistent control theory, good deal bounds, convexity corrections, and the dynamics of sovereign and bank CDS spreads. Agatha obtained her PhD in Mathematical Finance from the Stockholm School of Economics, under the supervision of Tomas Björk.