Stochastic Simulation and Monte Carlo Methods

1. Front Matter

   Pages I-XVI

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2. Principles of Monte Carlo Methods

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • Carl Graham, Denis Talay

      Pages 3-11

   3. Strong Law of Large Numbers and Monte Carlo Methods

      • Carl Graham, Denis Talay

      Pages 13-35

   4. Non-asymptotic Error Estimates for Monte Carlo Methods

      • Carl Graham, Denis Talay

      Pages 37-63

3. Exact and Approximate Simulation of Markov Processes

   1. Front Matter

      Pages 65-65

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   2. Poisson Processes as Particular Markov Processes

      • Carl Graham, Denis Talay

      Pages 67-88

   3. Discrete-Space Markov Processes

      • Carl Graham, Denis Talay

      Pages 89-119

   4. Continuous-Space Markov Processes with Jumps

      • Carl Graham, Denis Talay

      Pages 121-153

   5. Discretization of Stochastic Differential Equations

      • Carl Graham, Denis Talay

      Pages 155-195

4. Variance Reduction, Girsanov’s Theorem, and Stochastic Algorithms

   1. Front Matter

      Pages 197-197

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   2. Variance Reduction and Stochastic Differential Equations

      • Carl Graham, Denis Talay

      Pages 199-212

   3. Stochastic Algorithms

      • Carl Graham, Denis Talay

      Pages 213-230

5. Back Matter

   Pages 231-260

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In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view.

The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.

• 60H10, 65U05, 65C05, 60J30, 60E7, 65R20
• Markov processes
• Monte Carlo methods
• Stochastic calculus
• Stochastic numerical methods
• quantitative finance

• École Polytechnique, CNRS, Centre de Mathématiques Appliquées, Palaiseau, France

  Carl Graham

• INRIA, Sophia Antipolis, France

  Denis Talay

Carl Graham is a CNRS researcher and Professeur chargé de cours (part-time associate professor) at the École Polytechnique and associate editor for Annals of Applied Probability. His main fields of research include stochastic processes, stochastic modelling and communication networks. 

Denis Talay is a senior researcher at Inria. He holds a part time research position at École Polytechnique where he had taught for 13 years. He is, or has been, an associate editor for many top journals in probability, numerical analysis, financial mathematics and scientific computing. He was the president of the French Applied Math. Society SMAI (2006-2009) and is now the Chair of its Scientific Council. His main fields of interest are stochastic modelling, numerical probability, stochastic analysis of partial differential equations and financial mathematics.

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