Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations

This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially discretizing the Kolmogorov eq...

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Bibliographic Details
Main Authors: Bou-Rabee, Nawaf, Vanden-Eijnden, Eric
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2018
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Random walks (Mathematics)
Stochastic differential equations
Stochastic differential equations > Numerical solutions
parabolic partial differential equation
Stochastic differential equations
stochastic simulation algorithm
discrete maximum principle
Markov jump process
invariant measure
Fokker-Planck equation
stochastic Lyapunov function
geometric ergodicity
Kolmogorov equation
non-symmetric diffusions
monotone operator
ISBN:9781470431815, 1470431815
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Table of Contents:
  • Introduction -- Algorithms -- Examples & Applications -- Analysis on Gridded State Spaces -- Analysis on Gridless State Spaces -- Tridiagonal Case -- Conclusion and Outlook
  • Cover -- Title page -- Chapter 1. Introduction -- 1.1. Motivation -- 1.2. Main Results -- 1.3. Relation to Other Works -- 1.4. Organization of Paper -- 1.5. Acknowledgements -- Chapter 2. Algorithms -- 2.1. Realizability Condition -- 2.2. Gridded vs Gridless State Spaces -- 2.3. Realizable Discretizations in 1D -- 2.4. Realizable Discretizations in 2D -- 2.5. Realizable Discretizations in nD -- 2.6. Scaling of Approximation with System Size -- 2.7. Generalization of Realizable Discretizations in nD -- 2.8. Weakly Diagonally Dominant Case -- Chapter 3. Examples &amp -- Applications -- 3.1. Introduction -- 3.2. Cubic Oscillator in 1D with Additive Noise -- 3.3. Asymptotic Analysis of Mean Holding Time -- 3.4. Adaptive Mesh Refinement in 1D -- 3.5. Log-normal Process in 1D with Multiplicative Noise -- 3.6. Cox-Ingersoll-Ross Process in 1D with Multiplicative Noise -- 3.7. SDEs in 2D with Additive Noise -- 3.8. Adaptive Mesh Refinement in 2D -- 3.9. Log-normal Process in 2D with Multiplicative Noise -- 3.10. Lotka-Volterra Process in 2D with Multiplicative Noise -- 3.11. Colloidal Cluster in 39D with Multiplicative Noise -- Chapter 4. Analysis on Gridded State Spaces -- 4.1. Assumptions -- 4.2. Stability by Stochastic Lyapunov Function -- 4.3. Properties of Realizations -- 4.4. Generator Accuracy -- 4.5. Global Error Analysis -- Chapter 5. Analysis on Gridless State Spaces -- 5.1. A Random Walk in a Random Environment -- 5.2. Generator Accuracy -- 5.3. Stability by Stochastic Lyapunov Function -- 5.4. Finite-Time Accuracy -- 5.5. Feller Property -- 5.6. Stationary Distribution Accuracy -- Chapter 6. Tridiagonal Case -- 6.1. Invariant Density -- 6.2. Stationary Density Accuracy -- 6.3. Exit Probability -- 6.4. Mean First Passage Time -- Chapter 7. Conclusion and Outlook -- Bibliography -- Back Cover