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Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2015Copyright date: ©2015Edition: 1st edDescription: 1 online resource (112 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470422783
Subject(s): Genre/Form: Additional physical formats: Print version:: Numerical Approximations of Stochastic Differential Equations with Non-Globally Lipschitz Continuous CoefficientsDDC classification:
  • 519.20000000000005
LOC classification:
  • QA274.23.H889 2015
Online resources:
Contents:
Cover -- Title page -- Chapter 1. Introduction -- 1.1. Notation -- Chapter 2. Integrability properties of approximation processes for SDEs -- 2.1. General discrete-time stochastic processes -- 2.2. Explicit approximation schemes -- 2.3. Implicit approximation schemes -- Chapter 3. Convergence properties of approximation processes for SDEs -- 3.1. Setting and assumptions -- 3.2. Consistency -- 3.3. Convergence in probability -- 3.4. Strong convergence -- 3.5. Weak convergence -- 3.6. Numerical schemes for SDEs -- Chapter 4. Examples of SDEs -- 4.1. Setting and assumptions -- 4.2. Stochastic van der Pol oscillator -- 4.3. Stochastic Duffing-van der Pol oscillator -- 4.4. Stochastic Lorenz equation -- 4.5. Stochastic Brusselator in the well-stirred case -- 4.6. Stochastic SIR model -- 4.7. Experimental psychology model -- 4.8. Scalar stochastic Ginzburg-Landau equation -- 4.9. Stochastic Lotka-Volterra equations -- 4.10. Volatility processes -- 4.11. Overdamped Langevin equation -- Bibliography -- Back Cover.
Summary: Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, the authors illustrate their results for several SDEs from finance, physics, biology and chemistry.
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Cover -- Title page -- Chapter 1. Introduction -- 1.1. Notation -- Chapter 2. Integrability properties of approximation processes for SDEs -- 2.1. General discrete-time stochastic processes -- 2.2. Explicit approximation schemes -- 2.3. Implicit approximation schemes -- Chapter 3. Convergence properties of approximation processes for SDEs -- 3.1. Setting and assumptions -- 3.2. Consistency -- 3.3. Convergence in probability -- 3.4. Strong convergence -- 3.5. Weak convergence -- 3.6. Numerical schemes for SDEs -- Chapter 4. Examples of SDEs -- 4.1. Setting and assumptions -- 4.2. Stochastic van der Pol oscillator -- 4.3. Stochastic Duffing-van der Pol oscillator -- 4.4. Stochastic Lorenz equation -- 4.5. Stochastic Brusselator in the well-stirred case -- 4.6. Stochastic SIR model -- 4.7. Experimental psychology model -- 4.8. Scalar stochastic Ginzburg-Landau equation -- 4.9. Stochastic Lotka-Volterra equations -- 4.10. Volatility processes -- 4.11. Overdamped Langevin equation -- Bibliography -- Back Cover.

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, the authors illustrate their results for several SDEs from finance, physics, biology and chemistry.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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