但是我们会看到有一些stochastic process既是Gaussian又是Markov。 Brownian motion 如上图所述,Brownian motion可以看做是一种特殊粒子运动中位置随时间的变化X(t)。(当然也可以看作是某个股票价格随时间的变化。)我们通过特殊的方式引入随机性。将0到t时间分成n小段Δt=t/n,然后每过Δt粒子的位置可以加上Δx或...
We study the underdamped Brownian motion of a particle in a force field, with a stochastic force whose memory is short compared with the relaxation time, by the method of the stochastic Liouville equation. We show that such a system may be modelled approximately by a diffusion Markov process,...
Keywords Brownian motion martingale stochastic integral stochastic calculus Itô's formula martingale representation Markov process harmonic function stochastic differential equation quantitative finance Search within this book Search Table of contents (10 chapters) Front Matter Pages i-xiii ...
6 Brownian motion as a Markov processWe have seen in 2.13 that for a𝑑-dimensional Brownian motion(𝐵𝑡)𝑡⩾0and any𝑠>0the shifted process𝑊𝑡:= 𝐵𝑡+𝑠−𝐵𝑠,𝑡⩾0, is again a BM𝑑which is independent of(𝐵𝑡)0⩽𝑡⩽𝑠. Since𝐵𝑡+𝑠=𝑊...
1.3NondifferentiabilityofBrownianmotion18 1.4TheCameron–Martintheorem24 Exercises30 Notesandcomments33 2BrownianmotionasastrongMarkovprocess36 2.1TheMarkovpropertyandBlumenthal’s0-1law36 2.2ThestrongMarkovpropertyandthereflectionprinciple40 2.3MarkovprocessesderivedfromBrownianmotion48 2.4ThemartingalepropertyofBrow...
Process {xt}, t≥ 0, is a (scalar) constant-diffusion Brownian motion process if (a) {xt} is a process with independent increments and (b) for any nonnegative t1 and t2, the increments xt1−xt2 are Gaussian random variables with E(xt1−xt2)=0 and Ext1−xt22=qt1−t2,where...
. 10.2 Brownian motion as a Markov process Brownian motion enjoys the Markov property, the strong Markov property, and its transition density function, called the Gaussian kernel, satis?es a certain partial di?erential equation (PDE). 2 First, the collection {Ft}t∈IR+ is called the Brownian...
Preliminaries.§- Introduction.§- Martingales.§- Markov Processes.§- Stochastic Integration.§- Representation of Martingales.§- Local Times.§- Genera... D Revuz,M Yor - 世界图书出版公司 被引量: 6614发表: 2008年 On the representation of fractional Brownian motion as an integral with respec...
(BM). It unites in a remarkable synthesis intuitive andvisual simplicity with deep structural complexity and almost universal practical appli-cability. No attempt is made here even to review the subject. The aim is merely to setnecessary foundations for the future analysis of Markov processes. As...
In this article we define Brownian Motion and outline some of its properties, many of which will be useful when beginning to model asset price paths.In a previous article on the site we have introduced stochastic calculus in the context of its role in quantitative finance. The Markov and ...