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

Includes bibliography and index

Introduction: brief overview of Monte-Carlo methods
Part A: Toolbox for stochastic simulation
- Chapter 1- Generating random variables
- Chapter 2- Convergences and error estimates
- Chapter 3- Variance reduction
Part B: Simulation of linear processes
- Chapter 4- Stochastic differential equations and Feynman-Kac formulas
-Chapter 5- Euler scheme for stochastic differential equations
-Chapter 6- Statistical error in the simulation of stochastic differential equations
Part C: Simulation of non-linear processes
-Chapter 7- Backward stochastic differential equations
-Chapter 8- Simulation by empirical regression
-Chapter 9- Interacting particles and non-linear equations in the McKean sense
Appendis A- Reminders and complementary results
Bibliography
Index

The book focuses on the simulation of stochastic processes in continuous time and their link with partial differential equations (PDEs). It covers linear and nonlinear problems in biology, finance, geophysics, mechanics, chemistry, and other application areas. The text also thoroughly develops the problem of numerical integration and computation of expectation by the Monte-Carlo method.
It begins with a history of Monte-Carlo methods and an overview of three typical Monte-Carlo problems: numerical integration and computation of expectation, simulation of complex distributions, and stochastic optimization. The remainder of the text is organized in three parts of progressive difficulty. The first part presents basic tools for stochastic simulation and analysis of algorithm convergence. The second part describes Monte-Carlo methods for the simulation of stochastic differential equations. The final part discusses the simulation of non-linear dynamics.