Stochastic ordinary differential equations in applied and computational mathematics

Desmond Higham

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics, and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models, and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong
convergence properties are relevant for highly-efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on nonlinear models, parameter estimation, model comparison and multiscale simulation.
LanguageEnglish
Pages449-474
Number of pages26
JournalIMA Journal of Applied Mathematics
Volume76
Issue number3
DOIs
Publication statusPublished - Jun 2011

Fingerprint

Stochastic Ordinary Differential Equations
Ordinary differential equations
Multiscale Simulation
Diffusion Limit
Population dynamics
Multiscale Modeling
Model Comparison
Chemical Kinetics
Systems Biology
Stochastic models
Population Dynamics
Physical Model
Random processes
Reaction kinetics
Parameter estimation
Nonlinear Model
Stochastic Model
Parameter Estimation
Stochastic Processes
Numerical methods

Keywords

  • differential equations
  • computational mathematics
  • trajectories

Cite this

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Stochastic ordinary differential equations in applied and computational mathematics. / Higham, Desmond.

In: IMA Journal of Applied Mathematics, Vol. 76, No. 3, 06.2011, p. 449-474.

Research output: Contribution to journalArticle

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