# Newsletter of the Bachelier Finance Society

## Volume 11, Number 3, July 2019** **

**JOB POSTINGS**

The aim of these postings is to create a forum for the dissemination of information on academic and industrial positions related to mathematical finance, across different disciplines and different geographical regions. Please submit any job advertisements you are aware of to jobads@bachelierfinance.org, preferably in plain text and sending the link to the website containing all the information. Updates and new items appear continuously at: http://www.bachelierfinance.org/forum/jobs/.

**Research Internship Position**

NTU Singapore

**Lecturer**

University of Manchester

Deadline: July 12, 2019

**PhD positions **TU Berlin

Deadline: July 26, 2019

**Scientific Assistant**

HU Berlin

Deadline: July 26, 2019

**Scientific Assistant (part-time)**

HU Berlin

Deadline: July 26, 2019

**Professor position**

ETH Zurich

Deadline: September 30, 2019

**BOOKS & JOURNALS**

The Society maintains a list of books, book reviews and journals at: http://www.bachelierfinance.org/publications.html. Members that would like to have their books added to the website, should please let us know.

#### Recently published books

Gérard Cornuéjols, Javier Peña, Reha Tütüncü

**Optimization Methods in Finance,** 2nd Ed

Cambridge University Press (2018), ISBN 9781107056749

Marcos Lopez de Prado

**Advances in Financial Machine Learning**

John Wiley & Sons, Inc. (2018), ISBN-13: 978-1-119-48208-6

Roman Vershynin

**High-Dimensional Probability: An Introduction with Applications in Data Science**

Cambridge University Press (2018), ISBN 9781108415194

Lixin Wu

**Interest Rate Modeling: Theory and Practice, **2nd Ed.

Chapman & Hall/CRC (2019), ISBN 9780815378914

**BOOK REVIEW**

**Probabilistic Theory of Mean Field Games with Applications**

### by **René Carmona** and **François Delarue**

### reviewed by **Erhan Bayraktar**, University of Michigan

This two volume book (of 1410 pages) is a seminal text that gives a very comprehensive analysis of mean field games, *i.e.,* stochastic games with infinitely many small players *weakly* interacting through their aggregate distribution, introduced simultaneously by the foundational works of Lasry and Lions, and Huang, Malhame and Caines in 2006–2007. The advantage of the mean field formulation is that the equilibrium has an appealing decentralized structure: each player bases her decisions on her own state variable and the distribution of a population of exchangeable players, which one obtains from the solution of a fixed point problem. One can then use the mean field game solution to construct an approximate Nash equilibrium of the finite-player game.

Since the original articles, mean field games turned into a fertile area of research. This year the 5th conference on Mean Field Games and Related Topics will be held at CIRM, Italy in September 2019. Additionally, many conferences now have dedicated sessions on mean field games. There are many interesting research problems on both the mathematical side and in engineering, finance, economics and social sciences. This book makes an invaluable contribution by making the entry into this burgeoning field for new researchers, especially PhD students and post-docs, easier. Prior to this book the main references were the video lectures by Pierre-Louis Lions at the Collège de France (2007–2008), which are now translated into English, unpublished notes of Pierre Cardaliguet’s from these lectures available on his webpage since 2012, and the Springer Brief of Bensousan, Frehse and Yam in 2013. What was needed was a rigorous textbook covering the state of the art, and René Carmona and François Delarue filled this gap. The approach they use is analyzing Mean Field Games (MFGs) using Forward Backward Stochastic Differential equations (FBSDEs), an approach that goes back to the seminal paper of Carmona and Delarue in the SIAM Journal on Control and Optimization (SICON) in 2013.

The first volume is dedicated to mean field games without a common noise, *i.e.*, the dynamics of the state variable of each player are driven by independent idiosyncratic noises only. For pedagogical reasons the volume starts with a section on examples of mean field games (games of timing and a model of bank runs, Aiyagari’s growth model in economics, applications to systemic risk and order book models, and information percolation, just to name a few). The solutions to these examples are provided throughout the book. Section 2 is devoted to the analysis of games with finitely many players. Of particular interest is the linear-quadratic games which can be explicitly solved (and hence provides a test-bed for the theory to come). Sections 3 and 4 are the core of the first Part of Volume I in which the authors give general existence and uniqueness results. To solve a mean field game, one first determines the best response of a given player using standard stochastic control and then finds a fixed point in the space of flows of probability measures. The authors’ tool of choice for performing this analysis is FBSDEs as already pointed out. There are two distinct ways to formulate the problem with FBSDEs: The first way is to represent the value function using FBSDEs, and the second by using the Pontryagin maximum principle. Either way the FBSDEs that appear are of McKean–Vlasov kind, *i.e.*, the distribution of the forward component appears in the coefficients of both the forward and the backward components.

The second part of the first volume gives an analysis of the differential calculus over the the space of probability measures, which culminates with the Itô’s formula along the flow of measures. At the very start one finds treasures like the Glivenko–Cantelli Convergence Theorem with an optimal rate and its entire proof. The fifth chapter introduces the so-called *L*-derivative (named after Lasry and Lions) by a lifting procedure (from the metric space to a Hilbert space) and its relationship to other notions of derivatives that already exist on Wasserstein spaces defined in the theory of optimal transport. The so-called master equation, which is the decoupling field of the McKean–Vlasov SDEs, a partial differential equations (PDEs) on the space of probability measures, is introduced here, whetting the reader’s appetite for what is to come. The sixth chapter, in which controlled McKean–Vlasov equations are analyzed, is where some of these results are applied already. As the two problems can be easily confused, the authors emphasize the difference between controlling a McKean–Vlasov SDE and the mean-field games: one finds the fixed point first and then optimizes in the controlled McKean–Vlasov equations as opposed to optimizing first and then finding the fixed point. The controlled McKean–Vlasov problems are in fact connected to a problem in which a social planner coordinates a large population of exchangeable players. But an important class of games called potential mean field games, can actually be formulated as controlled McKean–Vlasov equations as the authors point out. The tool of choice in this chapter is the Pontryagin maximum principle. This is true even when they choose the usual alternative approach which relies on the dynamic programming principle (DPP). This is because they use the Pontryagin maximum principle to determine the optimal control in the feedback from. This then leads again to some convexity restrictions on the Hamiltonian, which one generally does not need when one uses the DPP approach. These restrictions can be avoided by proving the dynamic programming principle for open loop controls (see Bayraktar, Pham and Cosso’s paper in the Transactions of the American Mathematical Society in 2018, and the paper by Cosso and Pham that is to appear in Journal de Mathématiques Pures et Appliquées which analyzes the zero-sum games in this context). This chapter also has a short section which relates the optimal transport problem to controlled McKean–Vlasov dynamics. This link might prove to be quite useful as it might make transfer of technology between the two subjects possible. In fact, there are now some more results by Acciao, Backhoff and Carmona in arXiv on this connection, which would be nice to include in future editions.

The first volume ends with an epilogue which discusses several extensions which enlarge the range of applications: infinite horizon mean field games, ergodic mean field games, mean field games with finite state spaces. There is a very intriguing example reminiscent of the repeated Prisoner’s dilemma that illustrates that the limit of *N*-player games may not converge to the solution of the mean field game. These type of convergence results are visited in more generality in the second volume, which will be described next.

The first three chapters of the second volume, which comprises Part I, is dedicated to mean field games with common noise, an important generalization (since many applications dictate some correlation between the agents) in which the state dynamics of each agent is not only driven by its own idiosyncratic noise but also by a Brownian motion that is common to each agent. The results here are quite technical because the Schauder fixed point theorem does not apply since the space of solutions is too big. A discretization of the common noise becomes necessary and the resulting analysis takes some 200 pages: First, a whole new theory of FBSDEs in a random environment are developed in Chapter 1. Mean Field Games with common noise are introduced in Chapter 2, where strong and weak form of solutions are discussed and these are related to FBSDEs of the conditional McKean–Vlasov type. This part culminates with Chapter 3, where the authors approximate the problem when the conditioning is discrete and then extract a sequence by mean of tightness arguments. The authors provide some new material here: Although they had proved these results using relaxed controls in their paper with Lacker from 2016 in the Annals of Probability, the authors here stick to their tool of choice, FBSDEs. This preserves the consistency of the book and as a reader one appreciates the tremendous effort the authors must have put in. Part of Chapter 3 is dedicated to the discussion of uniqueness. Besides the usual Lasry–Lions monotonicity condition, the authors observe that in some cases uniqueness holds without this condition just due to the presence of the common noise. This phenomenon is quite related to the Peano phenomenon in the ODEs. As a reader one hopes that future editions of the book address this more since Delarue now has some nice results in this direction.

The second part of the book is related to the so-called master equation, infinite dimensional PDEs on the Wasserstein space of probability measures, which are the decoupling fields of the FBSDEs that appeared so far. PDEs on the Wasserstein space appeared before in the works of Ambrosio et al. and Otto et al. in the context of optimal transport but the PDEs that appear in Mean Field Games are non-linear and most importantly non-local. When there is no common noise the PDE is first order in the measure, when there is common noise the PDE in the measure argument becomes second order. In Chapter 4, they show by using the Dynamic Programming Principle and Itô’s formula for conditional flows developed in this chapter that the master field is a viscosity solution of the so-called master equation. This result appears for the first time in this book. A discussion on the uniqueness or comparison would have been much appreciated here: Proof of a comparison principle for second-order equations remains a challenging issue, except for the first order case which applies for deterministic dynamics, analyzed by Gangbo and Swiech in 2015 in Calculus of Variations. Another intriguing related observation concerns the definition of the viscosity solution: It can either be formulated through the test functions on a Hilbert space (as in the works of Pham and Wei in SICON in 2017 and ESAIM: COCV in 2018) or the metric space as it is done here. It is not entirely clear how these two PDE formulations are related since the spaces of test functions for the two formulations are different, because the lifting procedure to go to the Hilbert space formulation is likely to destroy smoothness as discussed by Cosso and Pham in their article to appear in Journal de Mathématiques Pures et Appliquées.

Maybe the most interesting chapters of the book are Chapters 5 and 6 of the second volume. First, in Chapter 5, this analysis extends the results of Crisan, Delarue and Chassegneux (arXiv preprint from 2014 and an updated version in 2015 of 98 pages) to analyze the regularity of the master equation. The only other paper that performs the regularity analysis in the second order case is the paper by Cardaliguet, Delarue, Lasry and Lions, which will appear as 224 page lecture notes in the Annals of Mathematics Studies by Princeton University Press in August 2019. The strategy in the latter work is through working with a Hamilton-Jacobi-Belman SPDE coupled with a Fokker-Planck SPDE (which this book slightly touches upon in Chapter 2 of this volume) instead of McKean-Vlaov FBSDE approach of the book under review. The main use of the regularity analysis is showcased in Section 6, where it is shown that under the conditions that the master equation has a unique smooth solution, the limits if the finite player closed loop Nash equilibria converges to the unique solution of the mean field game. Here they also have new regularity results for the *N*-player system. One should also mention the papers by Lacker (Probability Theory and Related Fields, 2016) and Fischer (Annals of Applied Probability, 2017), which analyzed the limit of open loop equilibria. Therein the uniqueness of the mean field game solution is no longer needed. Their point is to prove that the support of any weak limit of the laws of the empirical distributions of the finite player game equilibria is included in the set of solutions to the limiting mean-field game. A current hot topic is the identification of this support. Other recent related results that appeared after the book was published are the fluctuations (and large deviations and concentration of measures) results by Delarue, Lacker and Ramanan which are to appear in the Electronic Journal of Probability (and Annals of Probability). In the finite state space case these were done by Bayraktar and Cohen (SICON, 2019) and Cecchin and Pellino (Stochastic Processes and Their Applications, to appear).

One should also point out here that the first part of Chapter 6 looks into the converse of the above: if one uses the mean-field limit solution then one actually obtains near Nash-equilibrium, which was the main motivating driver of the study in mean field games. Here for the common noise case the authors use the conditional propagation of chaos results developed earlier in this volume, in Chapter 2.

The books ends with a final chapter on extensions to the mean field games with a major player (not to be confused with the Stackelberg games in which the major player acts first) and the mean field games of timing — in addition to the abstract results due to Carmona, Delarue, and Lacker (to appear in Applied Mathematics and Optimization), an explicit example due to Nutz in SICON 2018 is showcased.

A chapter (or perhaps a volume, to make it a trilogy!) on the numerical analysis would be much desired but perhaps too much to ask from the authors. The only reason to make such a request would be because of their expertise in the subject and the recent results of Chassagneux, Crisan and Delarue in the Annals of Applied Probability in 2019. A book type resource of this nature is sorely lacking from the community. Linear-quadratic games provide a nice test-bed for the theory (and we also need additional other explicit examples), but these games can be analyzed without the heavy machinery developed here. So in the end one needs reliable numerical analysis for the theory to be useful. Although there are several approaches proposed, a natural one being the Picard iteration, not many references explicitly address the rate of convergence of the implemented version of the underlying scheme. Some recent results (*e.g.* the one above in the Annals of Applied Probability and one by Bayraktar, Cohen and Budhiraja in SICON in 2018) now address the convergence question, but something that outlines and describes all existing numerical techniques (there are many due to Achdou, Benamou, and his collaborators), performs speed of convergence analysis, and analyzes the computational efficiency would benefit the community at large and increase the reach of the theory to many other research communities and application areas.

It could also be nice to carry out (maybe this is a whole new book proposal as the one above suggestion might also be) the whole exercise outlined in the book for continuous time interacting Markov chains, which are more amenable to computations. The analysis of mean field games goes back to Gomes, Mohr and Souza’s article in Applied Mathematics and Optimization in 2013, and to Guéant in 2015 in the same journal and there are some sections on this in the chapters devoted to applications in this book. But these could be extended: For example for Markov chains the Master equation is finite dimensional and is more amenable to analysis. Not only would this make this appealing to a broader group of researchers who want to apply the results, but it would also pedagogically enhance the text since it is easier to understand the underlying principles.

To sum it up: Although on a very technical subject and long (if the proofs were not of smaller font, we would have well over 2000 pages in front of us!), this treatise on mean field games is a delight to read. In many places the book contains new material just to keep the approach consistent throughout. Several extensions of their research are performed here that could have easily been nice papers. It is written by two of the most prominent figures in the field and covers all one needs to know on the subject. I appreciated every detail of this book: For example, the amount of knowledge put forth in the Notes and Complements after each chapter simply wowed me. It provides a thorough and deep picture of the whole field and offers excellent references for many related facts in probability, stochastic analysis and control.

I am personally thankful for such a great resource: It deepened my own understanding. This book is an essential reading to anyone who wants to enjoy a journey in the magical land of mean-field games.

**UPCOMING CONFERENCES**

This list contains conferences related to mathematical finance that take place in the next three months. A full list is available at http://www.bachelierfinance.org/congresses/conferences.html. Please let us know of conferences we are not aware of and include a URL for the event.

**The 7th Asian Quantitative Finance Conference (AQFC)**

July 2–5, 2019

Hanoi, Vietnam

**Vine copulas and their applications**

July 8–9, 2019

Munich, Germany

**Summer School to the 9th International Conference on Lévy Processes**

July 8–12, 2019

Athens, Greece

**3rd International Conference on Computational Finance (ICCF2019)**

July 8–12, 2019

A Coruña, Spain

**Summer School “Equilibria in Financial Markets: General Equilibrium and Game Theoretic Perspective”**

July 8–12, 2019

Bielefeld, Germany

**41st Stochastic Processes and its Applications Conference**

July 8–12, 2019

Chicago IL, USA

**23rd International Congress on Insurance: Mathematics and Economics (IME 2019)**

July 10–12, 2019

Munich, Germany

**9th International Conference on Lévy Processes**

July 15–19, 2019

Samos, Greece

Summer School is organized in Athens, Greece, from July 8–12, 2019

**Quant Summit USA**

July 17–18, 2019

New York NY, USA

**29th Jyväskylä Summer School**

August 5–16, 2019

Jyväskylä, Finland

**Advanced Risk and Portfolio Management (ARPM) Bootcamp**

August 12–17, 2019

New York NY, USA

**CSA2019 – Conference in Stochastic Analysis and Applications**

August 26–30, 2019

Risør, Norway

**12th European Summer School in Financial Mathematics**

September 2–6, 2019

Padova, Italy

**Vienna Congress on Mathematical Finance – VCMF 2019**

September 9–11, 2019

Vienna, Austria

**VCMF Educational Workshop 2019**

September 12–13, 2019

Vienna, Austria

**Recent developments in dependence modelling with applications in finance and insurance – 6th edition**

September 16–17, 2019

Island of Agistri, Greece

**Conference on Systemic Risk and Financial Stability**

September 19–20, 2019

Freiburg, Germany

**Workshop on “Nonstandard Investment Choice”**

September 20, 2019

Cergy, France

**MAFIA – Mathematical Finance and Analysis**

September 20–21, 2019

New York NY, USA

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