Introduction to Monte-Carlo methods for transport and diffusion equations/ Bernard Lapeyre, Etienne Pardoux and Remi Sentis.

Format:

Book

Author/Creator:

Lapeyre, Bernard.

Contributor:

Pardoux, E. (Etienne), 1947-

Sentis, Rémi

Class of 1924 Book Fund.

Series:

Oxford texts in applied and engineering mathematics ; 6.

Oxford texts in applied and engineering mathematics ; 6

Language:

English

Subjects (All):

Monte Carlo method.

Heat equation--Numerical solutions.

Heat equation.

Transport theory.

Physical Description:

ix, 163 pages ; 24 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2002.

Summary:

Monte-Carlo methods is the generic term given to numerical methods that use sampling of random numbers. This text is aimed at graduate students in mathematics, physics, engineering, economics, finance and the biosciences that are interested in using Monte-Carlo methods for the resolution of partial differential equations, transport equations, the Boltzmann equation and the parabolic equations of diffusion. It includes applied examples, particularly in mathematical finance, along with discussion of the limits of the methods and description of specific techniques used in practice for each example.

Contents:

1 Monte-Carlo methods and integration 1

1.1 Revision of probability theory 1

1.2 Description of the Monte-Carlo method 2

1.3 Limits and convergence 4

1.3.1 Convergence theorems 4

1.3.2 Estimate of the variance of a calculation 5

1.3.3 Some significant examples 6

1.4 Methods to reduce the variance 9

1.5 Sequences with weak discrepancy 14

1.6 Simulation of random variables 18

1.6.1 Simulation of a uniform distribution on [0, 1] 18

1.6.2 Simulation of other random variables 19

2 Transport equations and processes 21

2.1 Revision of Markov processes 23

2.1.1 Semigroup associated with a Markov process 25

2.2 Transport processes with discrete velocities 26

2.2.1 Jump Markov processes 26

2.2.2 Transport processes 32

2.2.3 Infinitesimal generator of the associated semigroup 33

2.3 Associated Kolmogorov equations 34

2.3.1 Fokker-Planck equation 34

2.3.2 The retrograde Kolmogorov equation 36

2.3.3 Generalization 37

2.4 Convergence to a diffusion 39

2.4.1 The central limit theorem for a jump process 39

2.4.2 Convergence of a random evolution to a diffusion 42

2.4.3 Convergence of the associated Kolmogorov equations 44

2.5 The general transport process 45

2.5.1 Jump Markov process 45

2.5.2 Transport processes and associated Kolmogorov equations 47

2.6 Application to transport equations 50

3 The Monte-Carlo method for the transport equations 53

3.1 Principle of the adjoint Monte-Carlo method 53

3.2 Principle of the direct Monte-Carlo method 56

3.2.1 Description of the method 57

3.2.2 Link with particle methods 60

3.3 Boundary conditions 61

3.4 General scheme with time discretization 64

3.5 Evaluation of the mesh quantities 66

3.6 Stationary problems 69

3.6.1 General scheme 70

3.6.2 Evaluation of the mesh quantities 71

3.7 Limits of the method and generalizations 71

3.7.1 Limits of the method 71

3.7.2 Operator splitting 72

3.7.3 Generalization to nonlinear problems 73

3.7.4 Coupling with other numerical methods 74

3.8 Specific techniques 76

3.8.1 Grouping 76

3.8.2 Fictitious shock technique 78

3.9 Reduction of variance and importance functions 79

3.9.1 Angular bias 81

3.9.2 Source bias 82

3.9.3 Geometry splitting 82

3.10 An example of angular bias 83

3.11 Remarks about programming 84

3.11.1 Vectorization 84

3.11.2 Parallelization 85

4 The Monte-Carlo method for the Boltzmann equation 87

4.1 General points about the Boltzmann equations 88

4.2 Link with the principal equation 92

4.2.1 Principal equation and the Bird collision process 94

4.2.2 Propagation of chaos 95

4.2.3 Interpretation in terms of the Monte-Carlo method 98

4.3 Linear and symmetric methods 99

4.3.1 Short description of the linear Monte-Carlo method 100

4.3.2 Description of the symmetric Monte-Carlo methods 102

4.4 Implementation of the symmetric methods 104

4.5 Limits of Monte-Carlo methods 104

5 The Monte-Carlo method for diffusion equations 107

5.1 Brownian motion and partial differential equations 107

5.1.1 Brownian motion 107

5.1.2 Brownian motion and the heat equation 110

5.1.3 The Ito stochastic integral 112

5.1.4 Brownian motion and the Dirichlet problem 119

5.1.5 Feynman-Kac formula 121

5.2 Probabilistic representations and the diffusion process 123

5.2.1 Stochastic differential equations 123

5.2.2 Infinitesimal generator and diffusion 127

5.2.3 Diffusions and evolution problems 129

5.2.4 Stationary problems and diffusion 133

5.2.5 Diffusion and the Fokker-Planck equation 134

5.2.6 Applications in financial mathematics 138

5.3 Simulation of diffusion processes 142

5.3.1 The Euler scheme 143

5.3.2 The Milshtein scheme 144

5.4 Methods for variance reduction 148

5.4.1 Control variables and predicted representation 148

5.4.2 Example of the use of control variables 151

5.4.3 Importance function and Girsanov's theorem 152.

Notes:

Includes bibliographical references (pages [157]-161) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

0198525923

0198525931

OCLC:

249399110