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## Weak Convergence to Stochastic Integrals

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0030
- Subject:
- Economics and Finance, Econometrics

The main object of this final chapter is to prove an essential companion result to the FCLT, the convergence of certain normalized random sums to stochastic integrals with respect to Brownian motion. ... More

## STOCHASTIC DIFFERENTIAL EQUATIONS AND INTEGRALS

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0005
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter returns to a mathematical topic — that of stochastic differential equations and stochastic integrals — the meaning of which are essential for the interpretation of the Langevin equation ... More

## Stochastic Integration

*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/9780199657063.003.0005
- Subject:
- Mathematics, Probability / Statistics, Applied Mathematics

The Itô stochastic integral with respect to a Brownian motion is constructed, and its properties are shown in detail. The Itô formula is proved. Its applications include Lévy’s characterization of a ... More

## Stochastic Integrals

*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:
- 10.1093/0198775180.003.0003
- Subject:
- Economics and Finance, Financial Economics

This chapter introduces the main technical tool for arbitrage pricing, namely stochastic integrals.

## 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

## Stochastic Integrals

*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/9780198851615.003.0004
- Subject:
- Economics and Finance, Econometrics

We introduce the Wiener process, the Itô stochastic integral, and derive the Itô formula. The connection with martingale theory is discussed, and there are several worked-out examples

## Langevin treatment of the Fokker–Planck process

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the Langevin treatment of the Fokker–Planck process and diffusion. The form of Langevin equation used is different from the stochastic differential equation using Ito's ... More

## Basic Itô calculus

*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/9780199654109.003.0006
- Subject:
- Mathematics, Mathematical Physics

This chapter deals with the basic Itô calculus. Fundamental notions such as predictability, martingales and stopping times are introduced in the discrete case. The chapter then gives a short account ... More

## Weak Convergence to Stochastic Integrals

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 2021
- Published Online:
- November 2021
- ISBN:
- 9780192844507
- eISBN:
- 9780191927201
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192844507.003.0032
- Subject:
- Economics and Finance, Econometrics

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary ... More

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