Continuous-Time Random Walks for the Numerical Solution of Stochastic Differential Equations.
- 1st ed.
- 1 online resource (136 pages)
- Memoirs of the American Mathematical Society Series ; v.256 .
- Memoirs of the American Mathematical Society Series .
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 & -- 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.
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 equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of SDEs.
9781470449193
Stochastic differential equations-Numerical solutions. Random walks (Mathematics).