Chapter 4 Brownian motion calculus. FROM Klebaner, Fima C
Brown运动的极限定理
摘 要Brown 运动的极限定理(Limit Theorem of Brownian Motion)是概率论极限理论的一个重要分支，对Brown 运动以及与Brown 运动相关随机过程轨道的性质的研究是一个广泛研究的课题.本文目的是研究布朗运动增量在一定条件下的极限定理，推广了前人的一些主要结果.本文的研究内容组织如下：第一章为绪论，介绍了Brown 运动的有关发展历史及已有的研究结果. 第二章为预备知识，介绍了一些记号与基本概念.第三章至第四章为本文研究的主要结果.在第三章，我们研究了布朗运动增量在Hölder 范数下的C-R 型局部泛函极限定理.第四章，研究了Brown 运动增量在Hölder 范数下关于,..r p C q s -容度的收敛速率.第五章，对本文工作总结及展望.关键词：Brown 运动；Hölder 范数；容度；收敛速率；AbstractLimit Theorem of Brownian motion is an important branch of Probability Limit Theorem. The topic on the path properties of Brownian motion and its relative stochastic process is widely researched. The purpose of this paper is to study Limit Theorem of Brownian motion and increments of a Brownian sheet under certain conditions. Some important results of predecessors are extended and improved. The content of this paper is organized as follows:The first chapter is an introduction which presents the development history and the existing research results of Brownian motion.The second chapter is preliminary knowledge which introduces signs and some basic concepts.The main results can be seen from chapter three to chapter four. The third chapter studies the theorem of C-R local functional limit for increments of a Brownian motion in Hölder norm. The forth chapter discusses the convergence rate of increments of a Brownian motion about the capacity of ,..r p C q s -in Hölder norm.The fifth chapter summarizes the content of this paper and makes the prospect.Key words: Brownian motion; Hölder norm; Capacity; The rate of convergence;第一章引言§1.1 Brown运动发展过程及有关应用Brown运动(Brownian motion，简称BM)在数学学科上也可以称为维纳过程，Brown运动作为具有连续时间参数和连续状态空间的一个随机过程，是随机过程学科中的最简单的、最基本的、最常见的随机过程之一.许多随机过程可看成Brown运动的推广或者泛函.Brown运动作为物理现象，首先由英国生物学家Robert Brown在1827年观察花粉微粒在液面上的“无规则运动”而提出，后来才由德耳索作出了正确的定性分析.在1905年，爱恩斯坦首次对这种“无规则运动”现象的物理规律，建立了一种数学模型，这一模型的问世使这一理论有了明显的发展.最后在Smoluchowski，Fokker，Planck，Burger，Furth，Ublenbeck等著名学者的努力下，这方面的理论工作得以迅速发展起来了.但在数学方面，由于缺少精确描述，因而进展较为缓慢，一直到了1918年才由Wiener提出了在Brown运动空间上定义测度与积分的精确且严格的数学定义，定义表明了Brown运动是一种独立增量过程，是一个具有连续时间参数和连续状态空间的随机过程.它是随机过程中最简单，最重要的特例，许多不同类型的重要随机过程都可以看做它的泛函或某种意义下的推广，这些工作推动了Brown运动研究的快速发展，并逐渐令其渗透到概率论的各个分支中，使之成为现代概率理论的重要篇章.在当今迅猛发展的时期，伴随着科学技术的快速发展和普及，又特别是计算机科学的广大应用，对Brown运动性质的探讨和研究意义深远且已取得了质的飞跃，从应用角度来看，工程技术，经济管理等广泛领域中都有“噪声”与涨落现象存在，它们往往涉及Brown运动，也就需要Brown运动的理论；又由于Brown运动与热传导方程有密切联系，使它成为概率论与分析联系的重要纽带.目前，六十年代中以来发展起来的Brown运动的极限定理已广泛地出现在多个领域中，如物理学、经济数学、通信理论、金融学、与数理统计等等学科.比如最经典的，也是较为突出的贡献就是将Brown运动与股票价格行为联系在一起，进而建立起Brown运动的股票价格数学理论模型，这是二十一世纪的一项具有突破性的重要意义的创新课题，给历史翻开了崭新的一页.当代资本市场理论的核心假设之一是Brown运动假设，市场理论认为证券期货价格具有随机性上下波动的特性，因此对Brown运动性质的研究在现代金融数学中起举足轻重的作用.因此诸多专家和学者对Brown运动及其相关的轨道性质进行了深入的研究.可见，对Brown运动的性质进行深入研究意义非常重大，这不仅极大的丰富了概率论的知识体系，而且为其实际应用提供了强而有力的理论指导.Brown运动的极限定理已经成为概率论极限理论学科中的一个比较热门的研究课题.国内外许多的概率论工作者纷纷对Brown运动的轨道的极限性质进行了广泛深入地研究.一些与Brown 运动有关的随机过程，比如广义Brown 运动[1-2]、Guass 过程[3-4]、扩散过程[5-6]、稳定过程[7]等的极限性质亦被大量研究.对Brown 运动轨道性质的研究是Brown 运动的极限定理主要研究内容之一，比如研究Brown 运动在一定的假设条件下的连续模定理[8-9]、Brown 运动Strassen 重对数律[10-11]，或者Brown 运动增量有多大[12]以及增量有多小[13]的问题，还有一些泛函极限定理问题[14-15]等等.布朗运动的极限定理作为一门广泛研究的课题，随着人们的不断深入研究，将会有更多新结果.§1.2 与本文研究有关的结果对Brown 运动的轨道性质的研究是Brown 运动极限定理的重要内容，发展至今已有几十年的历史.在这几十年里，随着研究的不断深入，研究成果不断丰富和完善起来，对它的研究既深化和丰富了极限理论学科中经典理论的重要的基本结果，同时也开拓了对其他随机过程重对数律的研究.后来，对Brown 运动增量的极限定理也做了许多研究，其中最重要的内容是Csörgö-Révész 有关研究结果，我们介绍如下：设(){}0B B t t =≥;是d -维标准Brown 运动，记[][]()(){}0000d d C T f C T R f =∈=,,;;.设不减函数()():00u a ∞→∞,,，满足： (1) ()0u a u u ≤∈∞,,； (2)uua 非减； (3) ()log /limlog log u u u a u→∞=∞. 对1-维Brown 运动，Csörgö和Révész [12-13]得到如下结果. 如果u a 满足(1)、(2)，则得到()()00limsup sup sup 1u uu u t u a s a B t s B t β→∞≤≤-≤≤+-=， ..a s ， (1-2-1)与()()00liminf sup sup 1u uu u t u a s a B t s B t γ→∞≤≤-≤≤+-=， ..a s ， (1-2-2)其中1/2log 2log u u u u u a a β-⎛⎫= ⎪⎝⎭且1/22log 8log u u u u u a a γπ-⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭.若(3)也成立，则(1-2-1)与(1-2-2)的上下极限可换为极限.(1-2-1)泛函版本已被Révész [10]给出.Révész 结果如下：命题1.2.1 若条件(1)、(2)成立，则[]()0,1/limsup sup,0u u u t a u t u K β→∞∈-∆-=， ..a s (1-2-3)且对所有K ϕ∈，有[]()0,1/limsupinf,0u u t a u u t u βϕ∈-→∞∆-=， ..a s (1-2-4)其中：()()()()u t u s B ut a s B ut ∆=+-,，01t ≤≤，[]01s ∈,[](){}12'001d 1dK C t t ϕϕ=∈≤⎰,;后来，危启才将结果推广到Hölder 范数情形[15]，结论如下：命题1.2.2若条件(1)、(2)成立，则 []()0,1/l i m s u ps u p ,0u u u t a ut u K αβ→∞∈-∆-=，..a s (1-2-5) 且对任意K ϕ∈，有[]()0,1/limsupinf,0u u t a u u t u αβϕ∈-→∞∆-=， ..a s (1-2-6)若条件(3)成立，则有[]()0,1/lim inf,0u u u t a u t u αβϕ→∞∈-∆-=， ..a s (1-2-7)后来，高付清、王清华[16]研究了命题1.2.1的收敛速率，其结果如下.命题1.2.3 设条件(1)、(2)成立，则对任意K ϕ∈，()1'0d 1t t ϕ<⎰，有[]()()0,1/log liminf 2loginf ,u u u t a u u u ut u b a βϕϕ→∞∈-∆-=， ..a s ， (1-2-8) 其中()()1/212'0/21d d c b t t ϕϕ⎛⎫⎪= ⎪ ⎪-⎝⎭⎰，d c 是正常数，精确值见Ciesielski 和Taylor [17]. 更进一步，若条件(3)成立，则[]()()0,1/log lim 2loginf ,u u u t a u uu ut u b a βϕϕ→∞∈-∆-=， ..a s (1-2-9)特别在(1)、(2)成立时，()()1/200log log liminf infsup 1u uu u t u a s ad u u u a B t s B t c a →∞≤≤-≤≤⎛⎫ ⎪⎪+-= ⎪ ⎪⎝⎭， ..a s (1-2-10)若(3)成立，则()()1/200log log lim infsup 1u uu u t u a s ad u u u a B t s B t c a →∞≤≤-≤≤⎛⎫ ⎪⎪+-= ⎪ ⎪⎝⎭， ..a s (1-2-11)不久，高付清和王清华的研究结果被推广到Hölder 范数情形[18].命题 1.2.4如果u a 为定义于()0+∞,上的非减函数，且(1)、(2)成立，那么对任意f K ∈，且()1I f <，有[]()()10,1/l o g l i m i n f l o g i n f u u u t a uu u u t u fb f a ααβ-→∞∈-⎛⎫∆-= ⎪⎝⎭,， ..a s (1-2-12)其中()()()1/21k b f I f αα-⎛⎫= ⎪ ⎪-⎝⎭，()k α是一个正常数，精确值见Baldi 与Roynette [14].命题1.2.5 如果u a 为定义于()0+∞,上的非减函数，且 (1)、(2)与(3)成立，那么对任意f K ∈，()1I f <，有[]()()10,1/l o g l i m l o g i n f u u u t a uu u u t u fb f a ααβ-→∞∈-⎛⎫∆-= ⎪⎝⎭,， ..a s (1-2-13)其中()()()1/21k b f I f αα-⎛⎫= ⎪ ⎪-⎝⎭，()k α是正常数，精确值见Baldi 与Roynette [14].§1.3 本文的主要工作及其结构安排读研期间，在导师的指导和帮助之下，对Brown 运动的轨道的一些极限性质进行了大量的学习和探索.受到文献[15]、[16]、[22]等的启发，本文研究布朗运动的局部泛函极限定理，得到了布朗运动增量在Hölder 范数下的C-R 型局部泛函极限定理。
斯普林格数学研究生教材丛书
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Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic 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LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。
分数brown运运动驱动的非lipschitz随随机微分方程
分数Brown运动驱动的非Lipschitz随机微分方程引言随机微分方程是描述具有随机性质的物理、生物、金融等领域中的动力学系统的数学模型。
分数Brown运动是一种具有长程记忆特性的随机过程，常用于描述非平稳时间序列的行为。
本文将讨论分数Brown运动驱动的非Lipschitz随机微分方程，包括定义、性质和数值解法等内容。
定义随机微分方程随机微分方程描述了一个未知函数与一个随机过程之间的关系。
一般形式的随机微分方程可以写作：dX(t)=b(X(t),t)dt+σ(X(t),t)dW(t)其中，X(t)表示未知函数，b(X(t),t)和σ(X(t),t)分别表示确定性部分和随机部分的函数。
dW(t)表示标准Wiener过程（或布朗运动）的增量。
分数Brown运动分数Brown运动是一种具有长程记忆特性的随机过程，其定义如下：S H(t)=∫x(t)−x(s)|t−s|1+H +∞−∞ds其中，S H(t)表示分数Brown运动，x(t)表示随机过程。
H称为Hurst指数，描述了分数Brown运动的长程记忆特性。
当H=0.5时，分数Brown运动退化为标准布朗运动。
非Lipschitz随机微分方程非Lipschitz随机微分方程是一类具有非线性增长项且满足非Lipschitz条件的随机微分方程。
非Lipschitz条件要求在某些点上，增长项的斜率无界。
这种方程的解是局部唯一的，并且可能存在多个解。
性质局部存在性和唯一性对于非Lipschitz随机微分方程，存在一个关于初值的局部存在性和唯一性定理，称为扰动Lipschitz条件。
该定理指出，当初值位于某个固定的区间内时，方程的解存在且唯一。
长程记忆性分数Brown运动驱动的非Lipschitz随机微分方程具有长程记忆性。
由于分数Brown运动本身具有长程记忆特性，它引入方程的解中，导致解的行为与时间间隔的选择密切相关。
这种长程记忆性是非平稳时间序列的典型特征。
Chapter_4_Laws_of_Motion
can neglect reaction forces
Free Body Diagram
Must identify all the forces acting on the object of interest Choose an appropriate coordinate system If the free body diagram is incorrect, the solution will likely be incorrect
g is the acceleration due to gravity
g can also be found from the Law of Universal Gravitation
Newton’s Third Law
If object 1 and object 2 interact, the force exerted by object 1 on object 2 is equal in magnitude but opposite in direction to the force exerted by object 2 on object 1. F 12 F 21
The net force is defined as the vector sum of all the external forces exerted on the object
External and Internal Forces
External force
Any force that results from the interaction between the object and its environment Forces that originate within the object itself They cannot change the object’s velocity
stochastic calculus for fractional brownian motion and related processes附录
kH (t, u)dWu = CH Γ (1 + α)
(2)
R
α (I− 1(0,t) )(x)dWx
(see Lemma 1.1.3). Therefore, the first equality is evident, since
0 R t
(kH (t, u))2 x)α )2 dx +
k n
2H
2
.
C . n2
(B.0.12)
References
[AOPU00] Aase, K., Øksendal, B., Privault, N., Ubøe, J.: White noise generalization of the Clark-Haussmann-Ocone theorem with applications to mathematical finance. Finance Stoch., 4, 465–496 (2000) [AS96] Abry, P., Sellan, F.: The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation. Appl. Comp. Harmon. Analysis, 3, 377–383 (1996) [AS95] Adler, R.J.; Samorodnitsky, G.: Super fractional Brownian motion, fractional super Brownian motion and related self-similar (super) processes. Ann. Prob., 23, 743–766 (1995) [ALN01] Al` os, E., Le´ on, I.A., Nualart, D.: Stratonovich stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Taiwanesse J. Math., 5, 609–632 (2001) [AMN00] Al` os, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Stoch. Proc. Appl., 86, 121–139 (2000) [AMN01] Al` os, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Prob., 29, 766–801 (2001) [AN02] Al` os, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep., 75, 129–152 (2002) [And05] Androshchuk, T.: The approximation of stochastic integral w.r.t. fBm by the integrals w.r.t. absolutely continuous processes. Prob. Theory Math. Stat., 73, 11–20 (2005) [AM06] Androshchuk, T., Mishura Y.: Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics. Stochastics: Intern. J. Prob. Stoch. Proc., 78, 281–300 (2006) [AG03] Anh, V., Grecksch, W.: A fractional stochastic evolution equation driven by fractional Brownian motion. Monte Carlo Methods Appl. 9, 189–199 (2003)
Fractional Brownian motion(fbm)
Fractional Brownian motion980128Fractional Brownian motion of (Hurst) exponent 0 <H < 1 is the zero mean Gaussian process B H (t ) with following properties.1.E [B (t )] =02. B H (0) = 03.B H (t +δ) -B H (t ) is normally distributed N (0, σ|δ|H )4.B H (t ) has independent increments.5. E [B H (t )B H (s )] =σ2/2(|t |2H + |s |2H - |t -s |2H )From 5 it follows that Var [B H (t )] =σ2|t |2H .B H (t ) is non-stationary and does not admit a spec-trum in the usual sense. However, it is possible to attach to it an average spectrumB H (t ) is exactly self-similar, perfectly determined by H . There are several way to create an approximation of FBm. One way is shown below, using the Random Midpoint Displacement (RMD). The algorithm can be explained by following comments.1. Start with two end-points.2. Add one point in the middle of this two points, and displace it with a random term (which depends on H ).3.Add point between all exciting points and displace them with random terms in the same way,until the desired number of points have been achieved.References.1. Beran, J.,Statistics for Long-Memory Processes , Chapman & Hall, New York, 1994.2. Lau, C., W.,Traffic Characterization, Quality of Service and System Design in Multimedia Broadband Networks , Phd theisis, The University of Texas at Austin.Matlab Script.function x = FBm(Ndyad,H)% Input:S ω()σ2ω2H 1+-------------------=% Ndyad: output vector length equal to 2^Ndyad + 1.% H: Hurst parameter.% Output:% x: trace of FBm.if nargin < 2disp(' usage: x = FBm[ Datalength(dyad), H)');return;endN=2^Ndyad;x(1) = 0;x(N+1) = randn(1,1)*2^(Ndyad*H);s(1) = sqrt(1 - 2^(2*H - 2));for i = 1:Ns(i+1) = 2^(-H)*s(i);endfor i = 1:Ndyadfor j = 0:2^(i-1)-1k = 2^(Ndyad+1-i);displmnt = randn(1,1)*2^(H*Ndyad)*s(i+1);x((k*j + k*(j+1) + 1)/2) = (x(k*j+1) + x(k*(j+1)+1))/2 + displmnt;endendExample.“Create a FBm trace of dyadic length 1025 and H = 0.8.”x = FBm(10, .8),plot(x)020040060080010001200−3−2.5−2−1.5−1−0.500.5x 104。
4.布朗运动与伊藤公式
Application of Central Limit Them.
Consider limit S (t ) / t as ∆→ 0.
S (t ) 1 t − tk ∆ tk +1 − t ∆ = ∆ S k +1 + ∆ S k t t tk +1 − t k +1 1 t − tk k = ∑ Ri + ∆ ∑ Ri t ∆ i =1 i =1 t − tk = ∆ tk tk k tk − t tk +1 k +1 ∑ Ri + ∑ Ri → X t i =1 ∆ t (k + 1) i =1
input these values to the ori. equation. This completes the proof of the lemma.
Geometric Brownian Motion
St*+∆t − St* = σ ∆t R (ω ) δ =: * St By Taylor expansion
1 2 * 2 δ = ln(1 + δ ) + δ + O ∆S 2 neglecting the higher order terms of ∆t, we have * St +∆t 1 2 ln * = σ ∆t R(ω ) − σ ∆t St 2rownian Motion cont.
S k∆ (t ), t = tk , ∆ S (t ) = t − t ∆ t − t ∆ k S k +1 + k +1 S k , tk ≤ t ≤ tk +1. ∆ ∆
Brownian Motion
dSt/St = µ dt + σ dWt,
(10.2)
which has an explicit solution
St = S0 exp[(µ − σ2/2) t + σWt].
(10.3)
Can we verify the solution by differentiating (10.3)? If you do it superficially by using the NewtonLeibniz calculus, you would not get back to (10.2). The reason is that {Wt} is a very special stochastic process, arguably the most important one in probability theory, named Brownian motion.
The close connection between Brownian motion and random walks is one of the most important facts in probability theory. Let ξ1, ξ2, ... be iid random variables with mean 0 and variance 1. For positive integer n, define a continuous-time process W (n) = {Wt(n)}t≥0 by
布朗运动、伊藤引理、BS 公式(前篇)
布朗运动、伊藤引理、BS 公式（前篇）对量化投资感兴趣的人大概都听说过的 Black-Scholes 期权定价公式（又称 Black-Scholes-Merton 公式，下称 BS 公式）。
它大概是将数学中随机过程（stochastic process）的概念运用到实际金融产品中的最著名的一个例子。
美国华尔街的 Quant 职位面试中更是无一例外的会问到 BS 公式及其引申出来的相关问题，足见其地位。
然而黑天鹅之父纳西姆·塔雷伯（Nassim Nicholas Taleb，以《黑天鹅效应》一书闻名于世）却对它嗤之以鼻，更是写过一篇题为 Why we have never used the Black-Scholes-Merton option pricing formula（为什么我们从来不用BS期权定价公式）来抨击它。
诚然，BS 公式在投资实践中能够起到多大的作用见仁见智。
但我们想说的是，BS 公式仅仅是一结果，是随机分析（stochastic calculus）经过严谨的层层推演得到的产物。
透过现象看本质，它背后蕴含着强大的数学体系，使得我们可以运用随机过程对股价、期权价格以及其他衍生品价格进行量化建模。
掌握这套分析体系对于有志于在量化投资领域有所建树的人来说十分必要。
想要摸清楚这套随机分析体系并不容易。
如果你在搜索引擎上查询 BS 公式的推导体系，一定会看到诸如“布朗运动”、“伊藤引理”、“随机微分方程”这些概念。
它们都是这套分析体系中必不可少的组成部分，环环相扣，在随机分析的大框架下完美的联系在一起。
熟悉这套分析框架的人可以充分的感受到这些基本模块无缝的组合在一起所展示出来的数学的魅力。
而对于不熟悉它的人来说，这之中每一个概念都可能仿佛天书一般；即便是具有高等数学知识的人，想要很快的梳理出它们之间的逻辑联系也并不容易。
简单的说，（标准）布朗运动是一种最简单的连续随机过程，它是描述证券价格随机性的基本模型。
Fractional Brownian Motion
Fractional Brownian Motion
Nov. 18, 2008
11 / 19
Fractional Brownian Motion
Fractional Brownian Motion
A myriad of methods have been developed for integrating fBM. Two systems bear the most importance in finance. The fractional pathwise integral has the form defined by
The jump in the value of this portfolio in one time-step is d Also we find that =r dt = r(V − S )dt. (1)
follows the random walk d = dV − dS
=
µS
∂V 1 ∂2V ∂V + + σ 2 S 2 2 − µS S dt ∂S ∂t 2 ∂S +σS
Yangho Choi (The University of Connecticut)
Fractional Brownian Motion
Nov. 18, 2008
2 / 19
Introduction
Introduction
History Bachelier, L (1900): Arithmetic Brownian Motion dS = µdt + σdB Samuelson, P.A. (1964): Geometric Brownian Motion dS = µdt + σdB S Osborne, M.F.M. (1959): The use of independent and normally distributed random variable to model the logarithm of stock prices ln St+τ St ∼ N(0, σ 2 τ )
菲赫金哥尔茨微积分学教程英文版
The Feynman-Kac formula is a fundamental result in stochastic calculus. It connects the solution of the heat equation to the path integral formulation of quantum mechanics. This is an introduction to the Feynman-Kac formula with applications to stochastic differential equations and mathematical finance.The first part of the book gives a brief introduction to the theory of stochastic processes and introduces the basic concepts of Brownian motion and the Gaussian measure. It then presents the Feynman-Kac formula for the heat equation and discusses its connection to the Schrödinger equation in quantum mechanics.The second part of the book covers applications of the Feynman-Kac formula to stochastic differential equations. It discusses the connection between the Feynman-Kac formula and the backward stochastic differential equation, and shows how the formula can be used to solve certain types of PDEs.The third part of the book focuses on applications of the Feynman-Kac formula to mathematical finance. It discusses the connection between the formula and the Black-Scholes equation, and shows how it can be used to price financial derivatives and model risk-neutral valuation.Overall, this book provides a comprehensive introduction to the Feynman-Kac formula and its applications. It is suitable for graduate students and researchers in mathematics, physics, and finance who are interested in stochasticcalculus and its applications.。
Brownian Motion and Itos Lemma
<∞
The process g is adapted to the FtW -filtration. 2. We say that the process g belongs to the class L2 [0, t] for all t > 0.
a.cartea@ – p. 5
a.cartea@ – p. 10
1 ′′ 2 ∆f (S) = f (S)∆S + f (S)(∆S) 2! 1 ′′′ + f (S)(∆S)3 + · · · 3! 1 ′′ ′ = f (S)∆W + f (S)(∆W )2 2! 1 ′′′ + f (S)(∆W )3 + · · · 3! 1 ′′ ′ 1/2 = f (S)φ∆t + f (S)φ2 ∆t2/2 2! 1 ′′′ + f (S)φ3 ∆t3/2 + · · · 3!
∆f ≡ f (S + ∆S) − f (S) 1 ′′ 1 ′′′ ′ 2 = f (S)∆S + f (S)(∆S) + f (S)(∆S)3 + · · · . 2! 3! What happens if S is not deterministic? Assume that the infinitesimally change in S is such that dS = dW .
Lecture 3: Brownian Motion and Ito’s Lemma
´ Alvaro Cartea Birkbeck College, University of London MSc Finance, Finance and Commodities and Financial Engineering
(金融保险类)金融数学或者经济数学方向的书单
=。
=+这些书单里的书俺都下了电子书...想催眠的想打发时间的想提高英语阅读能力的可以问我要...改天我自个整理个我下的经济学的电子书单...不想老是在后面再附上自己的推荐...******************************************************1. Futures, Options and other derivatives--by John Hull.这本书不用多说了，买就是了。
不管是找工作还是senior quant都会用到。
John Hull 也是非常厉害的，各个方面都有开创性的成果。
现在Toronto Uni.2. Arbitrage theory in continuous time--by Tomas Bjork这本书非常适合数学/物理背景的人读，注重数学理论的培养。
本来我觉得也没什么，但是被公司老板大加赞扬后就改变看法了。
Bjork现在瑞典SSE。
3. Financial Calculus--Martin Baxter& Rennie非常薄但是elegant的一本书，1996年，算是比较早了，但是和Hull的那本书齐名。
也是聪聪的first book。
作者1现在野村证券伦敦（nomura)，作者2在美林伦敦(ml)，都是fix ed income。
4.Financial calculus for finance II--ShreveShreve的新书，非常elegant，非常仔细，数学完备，适合数学背景，但是比较厚，对于入门来说还是3好。
作者现在CMU纽约。
教授。
顶尖人物。
5。
Martingale methods in Financial modelling--Musiela & Rutkovski作者现在BNP(巴黎银行）和华沙理工？都是顶尖人物。
数学背景1. Brownian motion and stochastic calculus--Shreve& Karasatz如果想在这一行发paper或者搞研究的话，或者读phd, 这是必须的。
brownian_motions
Definition through symmetric random walks
Consider a sequence Y = {Yi , i ∈ N+ } of iid random variables, such that P {Yi = 1} = P {Yi = −1} = 1 . 2
Now consider the discrete time process S = {Sn , n ∈ N+ } where
√ Thus assume ∆t → 0 under the condition ∆x = c ∆t for some c > 0. In this case E [X (t )] → 0 and Var[X(t)] → c2 t, for all t ∈ R+ , and the process defined in this way is called standard Brownian motion (when c = 1). Because of the above construction, the following properties are trivial X (t ) is normally distributed with zero mean and variance c 2 t (by the CLT); X (t ) has independent increments (being the limiting case of a symmetric random walk); X (t ) has stationary increments (being the limiting case of a symmetric random walk).
随机过程的连续时间模型
随机过程的连续时间模型随机过程是描述随机现象随时间变化的数学模型。
在概率论和统计学中，随机过程通常分为两种类型：离散时间模型和连续时间模型。
本文将重点探讨随机过程的连续时间模型。
连续时间随机过程的基本概念连续时间随机过程是在连续时间轴上取值的随机变量的集合。
它通常用随机变量X(t)表示，其中t是时间参数。
在连续时间随机过程中，任意时刻t下的随机变量X(t)都是一个随机变量。
维纳过程维纳过程（Wiener process）是最基本的连续时间随机过程之一。
它具有以下特性：1.维纳过程在任意时间段上的增量是独立同分布的。
2.维纳过程在任意时间段上的增量服从正态分布。
3.维纳过程有无限小尺度性质。
维纳过程在金融数学、物理学等领域有着广泛的应用。
布朗运动布朗运动（Brownian motion）是维纳过程的一种具体形式，其增量服从均值为 0、方差为t−s的正态分布。
布朗运动常用于研究粒子在流体中的随机运动、金融市场波动等现象。
随机微分方程随机微分方程（Stochastic differential equation，SDE）是描述连续时间随机过程演化的数学工具。
一般形式为：dX(t)=a(t,X(t))dt+b(t,X(t))dW(t)其中，a(t,X(t))和b(t,X(t))是确定性函数，dW(t)表示布朗运动的微分。
随机微分方程在金融工程、生态学、化学动力学等领域具有广泛的应用。
应用案例1.金融市场建模：使用随机微分方程描述资产价格演化，对金融市场波动进行建模和预测。
2.生态系统动力学：研究生态系统中种群演化的概率过程，探讨种群数量的随机波动。
3.通信系统性能分析：通过随机过程模型分析通信信道的噪声特性，验证通信系统的性能指标。
以上仅为随机过程连续时间模型应用的部分案例，随机过程在各个学科都有着重要的作用。
总结随机过程的连续时间模型提供了描述随机演化过程的数学框架，包括维纳过程、布朗运动和随机微分方程等重要概念。
Brown运动的两种局部时之间的关系
Brown运动的两种局部时之间的关系杜海霞;潘燕玲;刘利敏【摘要】Brown运动是一个具有连续时间参数和连续状态空间的随机过程,有两种不同定义下的局部时,一种是P.Levy提出的“mesure du voisinage”的概念,也即Brown运动{Wt,Ft}t≥0的局部时Lt(x)=limε→01/4εmeas{0≤s≤t；|Wt-x|≤ε},t∈[0,∞),(A) x∈R.另一种是由游程理论定义的局部时lt(x),并给出这两种局部时之间的关系Lt(0)=(√)2/4lt(0).【期刊名称】《西安文理学院学报（自然科学版）》【年(卷),期】2013(016)001【总页数】3页(P36-38)【关键词】Brown运动;局部时;游程;Tanaka公式【作者】杜海霞;潘燕玲;刘利敏【作者单位】郑州师范学院数学与统计学院,郑州450044;郑州师范学院数学与统计学院,郑州450044;河南师范大学数学与信息科学学院,河南新乡453007【正文语种】中文【中图分类】O211.63Brown运动作为物理现象，首先由英国生物学家Brown于1827年观察花粉微粒在液面上的“无规则运动”而提出．Einstein对这种“无规则运动”作了物理分析，并首次提出了Brown运动的数学模型，这个模型给出了Brown运动应遵从的概率分布．Wiener、P．Levy等人进一步研究了Brown运动的轨道性质．Wiener 提出了在Brown运动轨道空间上定义测度与积分，从而形成了Wiener空间的概念．此后对Brown运动及其泛涵的研究深入发展，又逐渐渗透到概率论及数学分析的各个领域中去，成为现代概率论的重要基础．Brown运动是一个具有连续时间参数和连续状态空间的随机过程．事实上，它是这样的随机过程中最简单、最重要的特例．许多不同类型的重要随机过程都可以看作它的泛函或某种意义下的推广．它又是迄今我们研究最多、了解最清楚、性质最丰富的随机过程之一．它重要而基本的理论在文献［1］［2］［4］中给出了阐述．P．Levy 提出了“mesure du voisinage”的概念，也即是我们称之为的 Brown 运动的局部时其中是标准一维 Brown 运动，文献［2］中对Brown运动的局部时Lt(x)进行了分析，给出了Lt(x)的性质:Lt(x)是连续可加泛涵，且成立．同时也给出了 Brown 运动的 Tanaka 公式:文献［3］中对游程理论进行了阐述，给出了由游程理论定义的局部时lt(x)及其性质:lt(x)也是连续可加泛涵，对于Brown运动来说，lt(x)与Lt(x)有相同的Lebesgue－Stieltjes测度支撑，本文主要研究这两种不同定义下的局部时之间的关系．1 Brown运动的两种局部时之间的关系设Wt(x)表示始于x的Brown运动，∀x∈R，易见:简记为Wt．定义1 设{X(t)，t≥0}是随机过程，如果它的有限维分布是空间平移不变的，即:则称此过程为空间齐次的．由可知:Brown运动具有空间齐次性，且Wt(x)和x+Wt(0)是相同的．故下面只讨论始于0点的Brown运动，也即标准一维Brown运动．设．而 lt(x)是由游程理论定义的Brown 运动的局部时，则Lt(x)与lt(x)都是连续可加泛涵，且有相同的Lebesgue－Stieltjes测度支撑．由于支撑相同的可加泛涵只相差一常数因子，所以不妨设Lt(x)=clt(x)，其中c为常数．由于Brown运动的预解式其中为有界可测函数．一方面，不妨令λ =1，f(y)=|y|，则:另一方面，记 Lt(0)为 Lt，lt(0)为 lt．(由 Tanaka公式)=综上可知于是由文献［3］知则有如下定理:定理1注由Tanaka公式可知:若令z=0，则a．s．P0，∀z∈R．由于Wt(x)－x与Wt(0)相同，故在任意一点x的局部时，可由Brown运动的空间齐次性，相应的转化为在0点的局部时．2 结语本文主要给出了Brow运动的局部时Lt(x)与由游程理论定义的局部时lt(x)之间的关系，以此为基础，就可以展开对与Brow运动有关的其它扩散过程的游程结构的分析，可以尝试得到它们的Tanaka公式．［参考文献］［1］钱敏平，龚光鲁．随机过程论［M］．北京:北京大学出版社，2004．［2］ IOANNISK，STEVEN E．Shreve，brownian motion and stochastic calculus［M］．New York:Spring－ Verlag World Publishing corp，1990．［3］ BLUMENTHAL R M．Excursions of Markov Processes ［M］．Birkhauser Boston Inc，1991．［4］张波．应用随机过程［M］．北京:中国人民大学出版社，2001．［5］徐侃．关于Brow运动的定义［J］．湖北师范学院学报，2007(4):120－122．［6］吕芳．生灭过程的Ito游程理论［J］．数学的实践与认识，2010(4):18－21．。
Brown 运动在H¨older 范数下的拟必然Strassen重对数律的收敛速率
Brown 运动在H¨older 范数下的拟必然Strassen重对数律的收敛速率黎协锐;刘永宏【期刊名称】《数学杂志》【年(卷),期】2016(036)002【摘要】本文研究了Brown 运动的泛函极限问题。
利用Brown 运动在H¨older 范数下关于容度的大偏差与小偏差，获得了Brown 运动在H¨older 范数下的Strassen 型重对数律的拟必然收敛速率，推广了文[2]中的结果。
%In this paper, limit question of Brownian motion is investigated. By using large and small deviations for Brownian motion in the H¨older norm with respect to Cr,p-capacity, the quasi sure convergence rate of Strassen’s type functional law of the iterated logarithm for Brownian motion in H¨older norm with respect to Cr,p-capacity is derived, which generalizes the result in [2].【总页数】9页(P310-318)【作者】黎协锐;刘永宏【作者单位】广西财经学院信息与统计学院，广西南宁 530003;桂林电子科技大学数学与计算科学学院，广西桂林 541004【正文语种】中文【中图分类】O211.4【相关文献】1.Brown运动在Hölder范数下的拟必然局部Strassen重对数律 [J], 谢德悦;刘永宏;李晓彬2.Brown 运动在容度意义下的局部Strassen重对数律 [J], 刘永宏3.Brown运动增量在H(o)lder范数下的拟必然收敛速度 [J], 黎协锐; 刘永宏4.Brown运动增量拟必然局部Strassen重对数律 [J], 李丰兵; 刘永宏5.Brown运动连续模在H lder范数下的拟必然收敛速度(英文) [J], 李余辉;邹小维;刘永宏;谢德悦因版权原因，仅展示原文概要，查看原文内容请购买。
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Klebaner, Fima C
Chapter 4: Brownian motion calculus
Klebaner, Fima C., (2005) Chapter 4: Brownian motion calculus. FROM: Klebaner, Fima C, Introduction to stochastic calculus with applications, 2nd edition, 2. pp.91-121. London: Imperial College Press. [31]
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