James Davidson
- 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. ...
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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. Some preliminary theory is given relating to random functionals on C and processes in continuous time.Less
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. Some preliminary theory is given relating to random functionals on C and processes in continuous time.
Robert M. Mazo
- 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 ...
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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 introduced in the previous chapter. The Ito and Stratonovich definitions of stochastic integrals are given. The question of which definition should be used in a given physical problem is discussed. Several examples are outlined.Less
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 introduced in the previous chapter. The Ito and Stratonovich definitions of stochastic integrals are given. The question of which definition should be used in a given physical problem is discussed. Several examples are outlined.
Gopinath Kallianpur and P. Sundar
- 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 ...
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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 Brownian motion, the Burkhölder-Davis-Gundy inequality, and the martingale representation theorem. Next, local times and the Tanaka formula are discussed. The Girsanov theorem on change of measures is proved in the last section.Less
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 Brownian motion, the Burkhölder-Davis-Gundy inequality, and the martingale representation theorem. Next, local times and the Tanaka formula are discussed. The Girsanov theorem on change of measures is proved in the last section.
Tomas Björk
- 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.
This chapter introduces the main technical tool for arbitrage pricing, namely stochastic integrals.
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 ...
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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 martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.Less
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 martingales are discussed. The book proceeds to construct stochastic integrals, prove the Itô formula, derive several important applications of the formula such as the martingale representation theorem and the Burkhölder-Davis-Gundy inequality, and establish the Girsanov theorem on change of measures. Next, attention is focused on stochastic differential equations which arise in modeling physical phenomena, perturbed by random forces. Diffusion processes are solutions of stochastic differential equations and form the main theme of this book. After establishing the existence and uniqueness of strong solutions to stochastic differential equations, weak solutions and martingale problems posed by stochastic differential equations are studied in detail. The Stroock-Varadhan martingale problem is a powerful tool in solving stochastic differential equations and is discussed in a separate chapter. The connection between diffusion processes and partial differential equations is quite important and fruitful. Probabilistic representations of solutions of partial differential equations, and a derivation of the Kolmogorov forward and backward equations are provided. Gaussian solutions of stochastic differential equations, and Markov processes with jumps are presented in successive chapters. The final objective of the book consists in giving a careful treatment of the probabilistic behavior of diffusions such as existence and uniqueness of invariant measures, ergodic behavior, and large deviations principle in the presence of small noise.
Tomas Björk
- 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
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
Melvin Lax, Wei Cai, and Min Xu
- 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 ...
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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 calculus lemma. The transform of the Langevin equation obeys the ordinary calculus rule, hence can be easily performed and some misleadings can be avoided. The origin of the difference between this approach and that using Ito's lemma comes from the different definitions of the stochastic integral. This chapter also discusses drift velocity, an example with an exact solution, use of Langevin equation for a general random variable, extension of this equation to the multiple dimensional case, and means of products of random variables and noise source.Less
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 calculus lemma. The transform of the Langevin equation obeys the ordinary calculus rule, hence can be easily performed and some misleadings can be avoided. The origin of the difference between this approach and that using Ito's lemma comes from the different definitions of the stochastic integral. This chapter also discusses drift velocity, an example with an exact solution, use of Langevin equation for a general random variable, extension of this equation to the multiple dimensional case, and means of products of random variables and noise source.
Jacques Franchi and Yves Le Jan
- 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 ...
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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 of the necessary background on martingales and Brownian motion, and the chapter deals finally with the basic tools of Itô's calculus: the stochastic integral and the Itô change-of-variable formula.Less
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 of the necessary background on martingales and Brownian motion, and the chapter deals finally with the basic tools of Itô's calculus: the stochastic integral and the Itô change-of-variable formula.
James Davidson
- 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 ...
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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 theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.Less
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 theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.
