Gobet - Monte-Carlo methods and stochastic processes: from linear to non-linear

This book originates from a third-year course on Monte-Carlo methods at Ecole Polytechnique University Paris Saclay. A version in French, with less material, is published by Editions de lEcole Polytechnique.

In fact, several Monte-Carlo methods exist. They all use random simulations, but they can be quite different in their techniques and objectives. It is not possible to present and analyze in detail all the existing methods in only one course. For this reason, I have made two choices.

1. The core of the book is related to the simulation of stochastic processes in continuous time and their link with partial differential equations. Although this link between Brownian motion and the heat equation goes back a century, the computational probabilistic aspects developed later. Progress in data-processing in the 1980s facilitated the task of researchers and engineers to test the algorithms, to improve them, to develop new ones, to strive to numerically simulate complex systems, and to increase the accuracy or the speed of simulation.
   The Monte-Carlo methods for the simulation of stochastic differential equations have many applications in biology (Wright-Fisher model), finance (valuation of options), geophysics (porous media), random mechanics (moving solid under random forces), fluid mechanics (Navier-Stokes equation for vorticity), etc. In , we will study some related methods of simulation.
   Since the end of the 1990s, important progress has been made in the field of non-linear processes in connection with problems of control or modeling of interaction. This is connected to topical issues, in research and in applications (chemistry, ecology, economy, finance, neurosciences, material physics, etc); we devote gathers the basic tools of simulation and analysis of algorithm convergence.

2. ].
   The problem of numerical integration and computation of expectation EX by the Monte-Carlo method is developed in this book thoroughly. A focus is made on the case where X is a path functional of a stochastic process in continuous time, possibly with non-linear dynamics.

Synopsis. The book is organized in three parts of progressive difficulty.

• Part A: Toolbox for Stochastic Simulation. The practice of random simulation requires the ability to simulate appropriate random variables. It is studied in tackles the question of acceleration of convergence (variance reduction methods). We emphasize the methods of importance sampling, whose application to the evaluation of rare events is spectacular.
• Part B: Simulation of Linear Processes. , we study the related statistical errors, the methods of variance reduction, and the multi-level Monte-Carlo methods.
• Part C: Simulation of Non-Linear Processes. The objective of the last part is to study non-linear dynamics, a field which is currently developing quickly, with a focus on their simulation. We present three generic non-linearities: backward equation and control, branching process, and mean-fields interaction. In .

At the end of each chapter, we provide some exercises of a theoretical or programming nature. Solutions and complementary material are available on the website

http://montecarlo-polytechnique.blogspot.com

Throughout this book, we emphasize the main algorithms, the most are generally not taught at the level of a masters program, but we have made a quite significant pedagogical effort to demystify them and make them available (in a simplified but not denatured form) to masters students. Proofs of results are usually given, some only outlined, but we often choose the simplest presentation and we try to use the arguments requiring fewer mathematical prerequisites.

Nevertheless, this is a quite demanding textbook of applied mathematics, covering a broad spectrum of advanced and sometimes very modern tools of probabilities, statistics and partial differential equations, with systematic computational concerns regarding numerical efficiency. Moreover, we encourage readers to implement the algorithms in order to develop their own computational intuition: this is certainly an important skill for mastering and understanding the theory. Moreover, the reader should keep in mind that even if an algorithm converges theoretically quicker than another, it may be that its execution time is much longer, and that it is actually less efficient: thus, comparing a convergence speed or an error variance is not all that is required; computational time and memory requirements may be significant features, which can be assessed only by implementing the method on a computer.

Our presentation of algorithms assumes that the implementation is made sequentially on a machine with a single processor. It is clear that the implementation on parallel architecture could be performed alternatively, and this is also an active field of research.

Last, it is difficult to be very original on such a classic subject on Monte-Carlo method and this Ecole Polytechnique course took, as a starting point, that of my predecessors (in particular L. Elie, C. Graham, B. Lapeyre, D. Talay). I would like to thank my colleagues who have encouraged me to transform my lecture notes into a published book. I especially thank P. Del Moral, S. De Marco, M. Gubinelli, and B. Jourdain for their feedback on a first version of this book. Thank you also to U. Stazhynski for his assistance in the translation from the French version to the English one.

Emmanuel Gobet, Paris Saclay