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System Control and Rough Paths

Terry Lyons and Zhongmin Qian

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.001.0001
Subject:
Mathematics, Probability / Statistics

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled ... More


Stochastic Dynamic Bifurcations and Excitability

Nils Berglund and Barbara Gentz

in Stochastic Methods in Neuroscience

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199235070
eISBN:
9780191715778
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199235070.003.0003
Subject:
Mathematics, Biostatistics

Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to ... More


Numerical Simulations of SDEs and SPDEs From Neural Systems Using SDELab

Hasan Alzubaidi, Hagen Gilsing, and Tony Shardlow

in Stochastic Methods in Neuroscience

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199235070
eISBN:
9780191715778
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199235070.003.0012
Subject:
Mathematics, Biostatistics

Stochastic differential equations are an important class of models that allow for a time varying random forcing in standard deterministic differential equations. We introduce the Itô stochastic ... More


Stochastic Differential Equations

Gopinath Kallianpur and P. Sundar

in Stochastic Analysis and Diffusion Processes

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199657063
eISBN:
9780191781759
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199657063.003.0006
Subject:
Mathematics, Probability / Statistics, Applied Mathematics

Stochastic differential equations arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of ... More


The Stochastic Simulation Method

Heinz-Peter Breuer and Francesco Petruccione

in The Theory of Open Quantum Systems

Published in print:
2007
Published Online:
February 2010
ISBN:
9780199213900
eISBN:
9780191706349
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199213900.003.07
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The formulation of the dynamics of open quantum systems by means of stochastic processes in Hilbert space leads to efficient Monte–Carlo simulation techniques which are introduced and examined in ... More


Stochastic Analysis and Diffusion Processes

Gopinath Kallianpur and P Sundar

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199657063
eISBN:
9780191781759
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199657063.001.0001
Subject:
Mathematics, Probability / Statistics, Applied Mathematics

Starting with the construction of stochastic processes, the book introduces Brownian motion and martingales. After proving the Doob-Meyer decomposition, quadratic variation processes and local ... More


LIPSCHITZ PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0002
Subject:
Mathematics, Probability / Statistics

The theory of controlled systems is worked out in detail for the case where the driving stimulus or noise is a Lipschitz function. The Itô functional, which is the map from stimulus to response, is ... More


VECTOR FIELDS AND FLOW EQUATIONS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on ... More


Applications to Financial Series

Christian Gouriéroux and Alain Monfort

in Simulation-based Econometric Methods

Published in print:
1997
Published Online:
November 2003
ISBN:
9780198774754
eISBN:
9780191596339
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774753.003.0006
Subject:
Economics and Finance, Econometrics

Applications to estimation of Stochastic Differential Equations, Stochastic Volatility Models, and Factor Models are considered. The methods used are Indirect Inference, Simulated Moments (based on ... More


Langevin Equations

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0009
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

In this chapter, Langevin equations (or Ito stochastic differential equations, SDEs) are derived that are equivalent to Fokker–Planck equations for bosons and fermions. The approach involves ... More


PATH INTEGRATION ALONG ROUGH PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0005
Subject:
Mathematics, Probability / Statistics

This chapter establishes an integration theory for rough paths. The key result is the existence of the integral of a Lipschitz one-form against a rough path. In particular, this gives a change of ... More


UNIVERSAL LIMIT THEOREM

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0006
Subject:
Mathematics, Probability / Statistics

The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential ... More


Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

Daniel Thomas Gillespie and Effrosyni Seitaridou

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199664504
eISBN:
9780191748516
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199664504.001.0001
Subject:
Physics, Soft Matter / Biological Physics

Brownian diffusion is the motion of one or more solute molecules in a sea of very many, much smaller solvent molecules. Its importance today owes mainly to cellular chemistry, since Brownian ... More


Continuous Markov process theory

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion: An Introduction to the Standard Theoretical Models

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199664504
eISBN:
9780191748516
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199664504.003.0007
Subject:
Physics, Soft Matter / Biological Physics

Three years after Einstein published his groundbreaking 1905 paper on diffusion, a different analysis was published by Paul Langevin. Although it was not clearly appreciated at the time, Langevin's ... More


Spin Dynamics at Finite Temperature

Olle Eriksson, Anders Bergman, Lars Bergqvist, and Johan Hellsvik

in Atomistic Spin Dynamics: Foundations and Applications

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198788669
eISBN:
9780191830747
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788669.003.0005
Subject:
Physics, Atomic, Laser, and Optical Physics

In Chapter 4 we presented a microscopic mechanism behind the LL equation and its connection to ab-initio results, such as that provided by density functional theory. All the analysis of Chapter 4 was ... More


Brownian motions on groups of matrices

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion: An Introduction

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.003.0007
Subject:
Mathematics, Mathematical Physics

This chapter is devoted to (left and right) Brownian motions on groups of matrices, which the chapter constructs as solutions to linear stochastic differential equations. The chapter establishes in ... More


Differential Equations

Tomas Björk

in Arbitrage Theory in Continuous Time

Published in print:
1998
Published Online:
November 2003
ISBN:
9780198775188
eISBN:
9780191595981
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198775180.003.0004
Subject:
Economics and Finance, Financial Economics

Here the theory developed in the previous chapter is used to analyse stochastic differential equations. These will later on be used to describe the behaviour of asset price processes.


Stochastic Differential Equations

Tomas Björk

in Arbitrage Theory in Continuous Time

Published in print:
2019
Published Online:
February 2020
ISBN:
9780198851615
eISBN:
9780191886218
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198851615.003.0005
Subject:
Economics and Finance, Econometrics

In this chapter we introduce stochastic differential equations (SDEs) and discuss existence and uniqueness questions. The geometric and linear equations are studied in some detail and their most ... More


Gaussian Solutions

Gopinath Kallianpur and P. Sundar

in Stochastic Analysis and Diffusion Processes

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199657063
eISBN:
9780191781759
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199657063.003.0009
Subject:
Mathematics, Probability / Statistics, Applied Mathematics

Gaussian solutions of stochastic differential equations play a special role in linear filtering problems. After a discussion of Gohberg-Krein special factorization, its connection to Gaussian ... More


Stochastic Optimal Control

Tomas Björk

in Arbitrage Theory in Continuous Time

Published in print:
1998
Published Online:
November 2003
ISBN:
9780198775188
eISBN:
9780191595981
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198775180.003.0014
Subject:
Economics and Finance, Financial Economics

This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general ... More


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