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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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- no detail
- some detail
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