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

##### By: Gobet, Emmanuel.

Publisher: Boca Raton CRC Press 2016Description: xxv, 309 p.ISBN: 9781498746229.Subject(s): Stochastic processes | Monte Carlo method | Linear | Non linearDDC classification: 519.2 Summary: Features Covers a broad spectrum of advanced and modern tools of probability, statistics, and PDEs, along with systematic computational concerns regarding numerical efficiency Emphasizes the main algorithms and most important convergence phenomena Encourages students to implement the algorithms to improve their own computational intuition Presents simple proofs of results Provides simulation exercises in Python on the author’s website A solutions manual and figure slides are available upon qualifying course adoption. Summary Developed from the author’s course at the Ecole Polytechnique, Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear 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. The book 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. https://www.crcpress.com/Monte-Carlo-Methods-and-Stochastic-Processes-From-Linear-to-Non-Linear/Gobet/p/book/9781498746229Item type | Current location | Item location | Collection | Call number | Status | Date due | Barcode |
---|---|---|---|---|---|---|---|

Books | Vikram Sarabhai Library | Slot 1399 (0 Floor, East Wing) | Non-fiction | 519.2 G6M6 (Browse shelf) | Available | 193521 |

Table of Contents

Introduction: brief overview of monte-carlo methods

a little history: from the buffon needle to neutron transport

problem 1: numerical integration: quadrature, monte-carlo, and quasi monte-carlo methods

problem 2: simulation of complex distributions: metropolis-hastings algorithm, gibbs sampler

problem 3: stochastic optimization: simulated annealing and robbins-monro algorithm

Toolbox for stochastic simulation

generating random variables

pseudorandom number generator

generation of one-dimensional random variables

acceptance-rejection methods

other techniques for generating a random vector

exercises

Convergences and error estimates

law of large numbers

central limit theorem and consequences

other asymptotic controls

non-asymptotic estimates

exercises

Variance reduction

antithetic sampling

conditioning and stratification

control variates

importance sampling

exercises

Simulation of linear process

stochastic differential equations and feynman-kac formulas

the brownian motion

stochastic integral and itô formula

stochastic differential equations

probabilistic representations of partial differential equations: feynman-kac formulas

probabilistic formulas for the gradients

exercises

Euler scheme for stochastic differential equations

definition and simulation

strong convergence

weak convergence

simulation of stopped processes

exercises

Statistical error in the simulation of stochastic differential equations

asymptotic analysis: number of simulations and time step

non-asymptotic analysis of the statistical error in euler scheme

multi-level method

unbiased simulation using a randomized multi-level method

variance reduction methods

exercises

Simulation of nonlinear process

backward stochastic differential equations

examples

feynman-kac formulas

time discretisation and dynamic programming equation

other dynamic programming equations

another probabilistic representation via branching processes

exercises

Simulation by empirical regression

the difficulties of a naive approach

approximation of conditional expectations by least squares methods

application to the resolution of the dynamic programming equation by empirical regression

exercises

Interacting particles and non-linear equations in the mckean sense

heuristics

existence and uniqueness of non-linear diffusions

convergence of the system of interacting diffusions, propagation of chaos, simulation

Appendix: reminders and complementary results

about convergences

several useful inequalities

Index

Features

Covers a broad spectrum of advanced and modern tools of probability, statistics, and PDEs, along with systematic computational concerns regarding numerical efficiency

Emphasizes the main algorithms and most important convergence phenomena

Encourages students to implement the algorithms to improve their own computational intuition

Presents simple proofs of results

Provides simulation exercises in Python on the author’s website

A solutions manual and figure slides are available upon qualifying course adoption.

Summary

Developed from the author’s course at the Ecole Polytechnique, Monte-Carlo Methods and Stochastic Processes: From Linear to Non-Linear 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.

The book 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.

https://www.crcpress.com/Monte-Carlo-Methods-and-Stochastic-Processes-From-Linear-to-Non-Linear/Gobet/p/book/9781498746229

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