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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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