Introduction to Monte-Carlo Methods for

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 ofpartial 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 foreach example.This is the sixth volume in the Oxford Texts in Applied and Engineering Mathematics series, which includes texts based on taught courses that explain the mathematical or computational techniques required for the resolution of fundamental applied problems, from the undergraduate through to thegraduate level. Other books in the series include: Jordan and Smith: Nonlinear Ordinary Differential Equations: An introduction to Dynamical Systems; Sobey: Introduction to Interactive Boundary Layer Theory; Scott: Nonlinear Science: Emergence and Dynamics of Coherent Structures; Tayler:Mathematical Models in Applied Mechanics; Ram-Mohan: Finite Element and Boundary Element Applications in Quantum Mechanics; Elishakoff and Ren: Finite Element Methods for Structures with Large Stochastic Variations.

1 Monte-Carlo methods and integration

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1.1 Revision of probability theory

1

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1.2 Description of the Monte-Carlo method

2

(2)

1.3 Limits and convergence

4

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1.3.1 Convergence theorems

4

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1.3.2 Estimate of the variance of a calculation

5

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1.3.3 Some significant examples

6

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1.4 Methods to reduce the variance

9

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1.5 Sequences with weak discrepancy

14

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1.6 Simulation of random variables

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

18

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1.6.2 Simulation of other random variables

19

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1.7 Bibliographic comments

20

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2 Transport equations and processes

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2.1 Revision of Markov processes

23

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2.1.1 Semigroup associated with a Markov process

25

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2.2 Transport processes with discrete velocities

26

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2.2.1 Jump Markov processes

26

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2.2.2 Transport processes

32

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2.2.3 Infinitesimal generator of the associated semigroup

33

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2.3 Associated Kolmogorov equations

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2.3.1 Fokker-Planck equation

34

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2.3.2 The retrograde Kolmogorov equation

36

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2.3.3 Generalization

37

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2.4 Convergence to a diffusion

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2.4.1 The central limit theorem for a jump process

39

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2.4.2 Convergence of a random evolution to a diffusion

42

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2.4.3 Convergence of the associated Kolmogorov equations

44

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2.5 The general transport process

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2.5.1 Jump Markov process

45

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2.5.2 Transport processes and associated Kolmogorov equations

47

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2.6 Application to transport equations

50

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2.7 Bibliographic comments

52

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3 The Monte-Carlo method for the transport equations

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3.1 Principle of the adjoint Monte-Carlo method

53

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3.2 Principle of the direct Monte-Carlo method

56

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3.2.1 Description of the method

57

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3.2.2 Link with particle methods

60

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3.3 Boundary conditions

61

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3.4 General scheme with time discretization

64

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3.5 Evaluation of the mesh quantities

66

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3.6 Stationary problems

69

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3.6.1 General scheme

70

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3.6.2 Evaluation of the mesh quantities

71

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3.7 Limits of the method and generalizations

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3.7.1 Limits of the method

71

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3.7.2 Operator splitting

72

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3.7.3 Generalization to nonlinear problems

73

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3.7.4 Coupling with other numerical methods

74

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3.8 Specific techniques

76

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3.8.1 Grouping

76

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3.8.2 Fictitious shock technique

78

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3.9 Reduction of variance and importance functions

79

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3.9.1 Angular bias

81

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3.9.2 Source bias

82

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3.9.3 Geometry splitting

82

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3.10 An example of angular bias

83

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3.11 Remarks about programming

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3.11.1 Vectorization

84

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3.11.2 Parallelization

85

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3.12 Bibliographic comments and conclusions

86

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4 The Monte-Carlo method for the Boltzmann equation

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4.1 General points about the Boltzmann equations

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4.2 Link with the principal equation

92

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4.2.1 Principal equation and the Bird collision process

94

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4.2.2 Propagation of chaos

95

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4.2.3 Interpretation in terms of the Monte-Carlo method

98

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4.3 Linear and symmetric methods

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4.3.1 Short description of the linear Monte-Carlo method

100

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4.3.2 Description of the symmetric Monte-Carlo methods

102

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4.4 Implementation of the symmetric methods

104

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4.5 Limits of Monte-Carlo methods

104

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4.6 Bibliographic comments

105

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5 The Monte-Carlo method for diffusion equations

107

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5.1 Brownian motion and partial differential equations

107

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5.1.1 Brownian motion

107

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5.1.2 Brownian motion and the heat equation

110

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5.1.3 The Itô stochastic integral

112

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5.1.4 Brownian motion and the Dirichlet problem

119

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5.1.5 Feynman-Kac formula

121

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5.2 Probabilistic representations and the diffusion process

123

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5.2.1 Stochastic differential equations

123

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5.2.2 Infinitesimal generator and diffusion

127

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5.2.3 Diffusions and evolution problems

129

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5.2.4 Stationary problems and diffusion

133

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5.2.5 Diffusion and the Fokker-Planck equation

134

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5.2.6 Applications in financial mathematics

138

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5.3 Simulation of diffusion processes

142

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5.3.1 The Euler scheme

143

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5.3.2 The Milshtein scheme

144

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5.4 Methods for variance reduction

148

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5.4.1 Control variables and predicted representation

148

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5.4.2 Example of the use of control variables

151

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5.4.3 Importance function and Girsanov's theorem

152

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5.5 Bibliographic comments

156

(1)

References

157

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Index

162

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Introduction to Monte-Carlo Methods for Transport and Diffusion Equations

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