布朗運動及其應(yīng)用

1、隨機過程在金融領(lǐng)域的作用1240410114王穎淺談布朗運動在金融領(lǐng)域的應(yīng)用懸浮微粒永不停息地做無規(guī)則運動的現(xiàn)象叫做布朗運動 例如，在顯微鏡下觀察懸浮在水中的藤黃粉、花粉微粒，或在無風(fēng)情形觀察空氣中的煙粒、塵埃時都會看到這種運動。溫度越高，運動越激烈。它是1827年植物學(xué)家R.布朗首先發(fā)現(xiàn)的。作布朗運動的粒子非常微小，直徑約110微米， 在周圍液體或氣體分子的碰撞下，產(chǎn)生一種漲落不定的凈作用力，導(dǎo)致微粒的布朗運動。如果布朗粒子相互碰撞的機會很少，可以看成是巨大分子組成的理想氣體，則在重力場中達到熱平衡后，其數(shù)密度按高度的分布應(yīng)遵循玻耳茲曼分布。J.B.佩蘭的實驗證實了這一點，并由此相當精確地測

2、定了阿伏伽德羅常量及一系列與微粒有關(guān)的數(shù)據(jù)。1905年A.愛因斯坦根據(jù)擴散方程建立了布朗運動的統(tǒng)計理論。布朗運動的發(fā)現(xiàn)、實驗研究和理論分析間接地證實了分子的無規(guī)則熱運動，對于氣體動理論的建立以及確認物質(zhì)結(jié)構(gòu)的原子性具有重要意義，并且推動統(tǒng)計物理學(xué)特別是漲落理論的發(fā)展。由于布朗運動代表一種隨機漲落現(xiàn)象，它的理論對于儀表測量精度限制的研究以及高倍放大電訊電路中背景噪聲的研究等有廣泛應(yīng)用。 這是1826年英國植物學(xué)家布朗（1773-1858）用顯微鏡觀察懸浮在水中的花粉時發(fā)現(xiàn)的。后來把懸浮微粒的這種運動叫做布朗運動。不只是花粉和小炭粒，對于液體中各種不同的懸浮微粒，都可以觀察到布朗運動。 布朗的發(fā)現(xiàn)

3、是一個新奇的現(xiàn)象，它的原因是什么？人們是迷惑不解的。在布朗之后，這一問題一再被提出，為此有許多學(xué)者進行過長期的研究。一些早期的研究者簡單地把它歸結(jié)為熱或電等外界因素引起的。最早隱約指向合理解釋的是維納(18261896)，1863年他提出布朗運動起源于分子的振動，他還公布了首次對微粒速度與粒度關(guān)系的觀察結(jié)果。不過他的分子模型還不是現(xiàn)代的模型，他看到的實際上是微粒的位移，并不是振動。 到了7080年代，一些學(xué)者明確地把布朗運動歸結(jié)為液體分子撞擊微粒的結(jié)果，這些學(xué)者有卡蓬內(nèi)爾、德爾索和梯瑞昂，還有耐格里。植物學(xué)家耐格里(1879)從真菌、細菌等通過空氣傳播的現(xiàn)象，認為這些微粒即使在靜止的空氣中也可

4、以不沉。聯(lián)系到物理學(xué)中氣體分子以很高速度向各方向運動的結(jié)論，他推測在陽光下看到的飛舞的塵埃是氣體分子從各方向撞擊的結(jié)果。他說：“這些微小塵埃就象彈性球一樣被擲來擲去，結(jié)果如同分子本身一樣能保持長久的懸浮?！辈贿^耐格里又放棄了這一可能達到正確解釋的途徑，他計算了單個氣體分子和塵埃微粒發(fā)生彈性碰撞時微粒的速度，結(jié)果要比實際觀察到的小許多數(shù)量級，于是他認為由于氣體分子運動的無規(guī)則性，它們共同作用的結(jié)果不能使微粒達到觀察速度值，而在液體中則由于介質(zhì)和微粒的摩擦阻力和分子間的粘附力，分子運動的設(shè)想不能成為合適的解釋。 無規(guī)則行走（random walk）    定義：無規(guī)則行走就是隨

5、機游走。其概念接近于布朗運動，是布朗運動的理想數(shù)學(xué)狀態(tài)。 核心概念：任何無規(guī)則行走者所帶的守恒量都各自對應(yīng)著一個擴散運輸定律。 就這樣，布朗運動自發(fā)現(xiàn)之后，經(jīng)過多半個世紀的研究，人們逐漸接近對它的正確認識。到本世紀初，先是愛因斯坦和斯莫盧霍夫斯基的理論，然后是貝蘭和斯維德伯格的實驗使這一重大的科學(xué)問題得到圓滿地解決，并首次測定了阿伏加德羅常數(shù)，這也就是為分子的真實存在提供了一個直觀的、令人信服的證據(jù)，這對基礎(chǔ)科學(xué)和哲學(xué)有著巨大的意義。從這以后，科學(xué)上關(guān)于原子和分子真實性的爭論即告終結(jié)。正如原先原子論的主要反對者奧斯特瓦爾德所說：“布朗運動和動力學(xué)假說的一致，已經(jīng)被貝蘭十分圓滿地證實了，這就使那

6、怕最挑剔的科學(xué)家也得承認這是充滿空間的物質(zhì)的原子構(gòu)成的一個實驗證據(jù)”。數(shù)學(xué)家和物理學(xué)家彭加勒在1913年總結(jié)性地說道：“貝蘭對原子數(shù)目的光輝測定完成了原子論的勝利”?！盎瘜W(xué)家的原子論現(xiàn)在是一個真實存在“。布朗運動代表了一種隨機漲落現(xiàn)象，它的理論在其他領(lǐng)域也有重要應(yīng)用。如對測量儀器的精度限度的研究；高倍放大電訊電路中的背景噪聲的研究等 布朗運動與分子熱運動不一樣，與溫度和粒子個數(shù)有關(guān)，溫度越高，布朗運動越劇烈，粒子越少，分子熱運動越劇烈。分子永不停息地做無規(guī)則的運動：分子永不停息地做無規(guī)則的運動。布朗運動、擴散現(xiàn)象都說明了任何物質(zhì)的分子，不論在什么狀態(tài)下，都在做永不停息的無規(guī)則運動。分子的無規(guī)則

7、運動與物質(zhì)的溫度有關(guān)，溫度越高，分子的無規(guī)則運動越劇烈。 將布朗運動與股票價格行為聯(lián)系在一起，進而建立起維納過程的數(shù)學(xué)模型是本世紀的一項具有重要意義的金融創(chuàng)新，在現(xiàn)代金融數(shù)學(xué)中占有重要地位。迄今，普遍的觀點仍認為，股票市場是隨機波動的，隨機波動是股票市場最根本的特性，是股票市場的常態(tài)。 布朗運動假設(shè)是現(xiàn)代資本市場理論的核心假設(shè)?，F(xiàn)代資本市場理論認為證券期貨價格具有隨機性特征。這里的所謂隨機性，是指數(shù)據(jù)的無記憶性，即過去數(shù)據(jù)不構(gòu)成對未來數(shù)據(jù)的預(yù)測基礎(chǔ)。同時不會出現(xiàn)驚人相似的反復(fù)。隨機現(xiàn)象的數(shù)學(xué)定義是：在個別試驗中其結(jié)果呈現(xiàn)出不確定性；在大量重復(fù)試驗中其結(jié)果又具有統(tǒng)計規(guī)律性的現(xiàn)象。描述股價行為模型

8、之一的布朗運動之維納過程是馬爾科夫隨機過程的一種特殊形式；而馬爾科夫過程是一種特殊類型的隨機過程。隨機過程是建立在概率空間上的概率模型，被認為是概率論的動力學(xué)，即它的研究對象是隨時間演變的隨機現(xiàn)象。所以隨機行為是一種具有統(tǒng)計規(guī)律性的行為。股價行為模型通常用著名的維納過程來表達。假定股票價格遵循一般化的維納過程是很具誘惑力的，也就是說，它具有不變的期望漂移率和方差率。維納過程說明只有變量的當前值與未來的預(yù)測有關(guān)，變量過去的歷史和變量從過去到現(xiàn)在的演變方式則與未來的預(yù)測不相關(guān)。股價的馬爾科夫性質(zhì)與弱型市場有效性(the weak form of market efficiency)相一致，也就是說

9、，一種股票的現(xiàn)價已經(jīng)包含了所有信息，當然包括了所有過去的價格記錄。但是當人們開始采用分形理論研究金融市場時，發(fā)現(xiàn)它的運行并不遵循布朗運動，而是服從更為一般的分數(shù)布朗運動。布朗運動貫穿于金融領(lǐng)域中，在現(xiàn)代金融領(lǐng)域中占有重要地位。 布朗運動指的是一種無相關(guān)性的隨機行走，滿足統(tǒng)計自相似性,即具有隨機分形的特征，但其時間函數(shù)(運動軌跡)卻是自仿射的。可用函數(shù)表示：如()(0)Xtt£是標準布朗運動，則下列各個隨機函數(shù)也是標準布朗運動。 1）、21()(/)XtcXtc=    （c0為常數(shù)，t0） 2）、2()()()Xt

10、XthhX（h0為常數(shù)，t0） 3）、13()(0)()0(0)tXttXtt具有以下主要特性：粒子的運動由平移及其轉(zhuǎn)移所構(gòu)成，顯得非常沒規(guī)則而且其軌跡幾乎是處處沒有切線；粒子之移動顯然互不相關(guān)，甚至于當粒子互相接近至比其直徑小的距離時也是如此；粒子越小或液體粘性越低或溫度越高時，粒子的運動越活潑；粒子的成分及密度對其運動沒有影響；粒子的運動永不停止。 二布朗運動在金融領(lǐng)域的發(fā)展   1900年法國的巴施利葉在博士論文投機理論中將股票價格的漲跌也看作是一種隨機運動，所得到的方程與描述布朗粒子運動的方程非常相似。第一次給予布朗運動以嚴格的數(shù)學(xué)描述

11、。但由此得到的股票價格可能取負值，顯然與實際不符當然，巴施利葉所謂的“布朗運動”，實質(zhì)上指的是股市的價格變動，換句話說，他把股價的變動，理想化為布朗運動可見，在物理學(xué)界尚未把布朗運動研究清楚之前，它象征“無規(guī)行走 的意義，早就被經(jīng)濟研究所吸納了。控制論創(chuàng)始人維納于1923年對布朗運動作出了嚴格的數(shù)學(xué)定義，根據(jù)這一定義，布朗運動是一種獨立增量過程，因而是一種馬爾科夫過程，數(shù)學(xué)界也常把布朗運動稱為維納過程。Markowiz(1952)發(fā)表投資組合選擇理論；Roberts和Osborne(1959)把隨機數(shù)游走和布朗運動的概念帶入股市研究；Samuelson和Fama(1970)的有效市場

12、理論(EMH)；Fischer Black和（Black-Scholes模型）；Ross (1976)的套利定價理 .布朗運動假設(shè)是現(xiàn)代資本市場理論的核心假設(shè)?，F(xiàn)代資本市場理論認為證券期貨價格具有隨機性特征。這里的所謂隨機性，是指數(shù)據(jù)的無記憶性，即過去數(shù)據(jù)不構(gòu)成對未來數(shù)據(jù)的預(yù)測基礎(chǔ)。同時不會出現(xiàn)驚人相似的反復(fù)。股價行為模型通常用著名的維納過程來表達。假定股票價格遵循一般化的維納過程是很具誘惑力的，也就是說，它具有不變的期望漂移率和方差率。  將布朗運動與股票價格行為聯(lián)系在一起，進而建立起維納過程的數(shù)學(xué)模型是本世紀的一項具有重要意義的金融創(chuàng)新，在

13、現(xiàn)代金融數(shù)學(xué)中占有重要地位。迄今，普遍的觀點仍認為，股票市場是隨機波動的。隨機波動是股票市場最根本的特性，是股票市場的常態(tài)。布朗運動假設(shè)是現(xiàn)代資本市場理論的核心假設(shè)。現(xiàn)代資本市場理論認為證券期貨價格具有隨機性特征。這里的所謂隨機性，是指數(shù)據(jù)的無記憶性，即過去數(shù)據(jù)不構(gòu)成對未來數(shù)據(jù)的預(yù)測基礎(chǔ);同時不會出現(xiàn)驚人相似的反復(fù)。英文翻譯：On the application of Brown in the financial fieldThe phenomenon of suspended particles never cease to do no regular movement is called

14、Brown.For example, the observation of gamboge powder suspended in the water under the microscope, pollen grains, or in case of no observation of smoke in the air dust particle, will see this movement. The higher the temperature is, the more intense exercise. It is 1827 botanist R. Brown first discov

15、ered. Brown motion of the particle diameter is very small. 110 um, in the collision of surrounding liquid or gas molecules, the net force generated an irregular movement, Brown lead particles. If Brown particles collide with each other little chance can be regarded as the ideal gas giant molecules,

16、then reach the heat balance in the gravity field, its number density according to the height of The distribution should follow the Boltzmann distribution.J.B. Perrin experiments confirmed this point, and thus quite accurately measured A Fugadero constant and a series of related data of.1905 and part

17、icle diffusion equation is established according to the Einstein A. statistical theory of Brown motion. Brown movement, experimental study and theoretical analysis indirectly confirmed no thermal motion of molecules, the kinetic theory of gases and confirm the atomic structure of the material is of

18、great significance, and promote the development of statistical physics, especially the fluctuation theory. Due to the movement of Brown represents a random fluctuation, its theory for measuring precision Study on the limit and high magnification of background noise in the circuit of telecommunicatio

19、ns is widely used. This is the 1826 British botanist Brown (1773-1858) observed with a microscope found in the water suspension of the pollen. The motion of suspended particles and later called the Brown movement. Not only the pollen and small carbon particles, suspended particles for liquid in diff

20、erent, we can observe the movement of Brown.Brown's discovery is a new phenomenon, what is it? It is very much puzzled. After Brown, this problem has been put forward, so many scholars have conducted long-term research. Some early researchers simply put it down to heat or electricity and other e

21、xternal factors caused the most. A reasonable explanation is early to Wiener (1826 - 1896), in 1863 he proposed the Brown vibration movement originated in the molecule, he published observations on the particle velocity and size relations for the first time. But his molecular models are not modern m

22、odels, he saw was actually the particle displacement, and no vibration.In 70 - 80 years, some scholars consider Brown explicitly attributed to liquid molecular collision particle results, these scholars have Capon Neil Delsol and rayon and ladder, Negri. Negri (1879) botanist from fungi, bacteria sp

23、read through the air, that these particles even in still air or not heavy. Linked to the gas molecular physics movement in all directions at a high speed of the conclusion, he speculated that the flying dust is gas molecules from the direction of impact results seen in the sun. He said: "being

24、thrown around these tiny dust like elastic ball, the molecule itself as Can keep a long suspension. "But this may Negri and give up the way to the correct interpretation, he calculated the individual gas molecules and dust particles velocity particle elastic collision, much smaller than the act

25、ual magnitude results observed, so he believes that because of the irregularity of the movement of gas molecules, they the result of the interaction can make the particles reach observation value of velocity in liquid due to friction and adhesion between particles and medium molecular, molecular mot

26、ion assumption can not be the appropriate interpretation.Irregular random (walk)Definition: the random walk is a random walk. The concept is close to the Brown movement, which is the ideal state of Brown's motion.The core concept: the conserved quantities of any irregular walkers are correspondi

27、ng to a diffusive transport law.It is important to establish the mathematical model of the Wiener process, which is an important financial innovation in modern financial mathematics. So far, the stock market is the most fundamental characteristic of the stock market.Brown's hypothesis is the cor

28、e of modern capital market theory. The modern capital market theory holds that stock futures price is random, which means that the data is non memory, that is, the prediction of the past data is not the basis of the future data. The mathematical definition of the random phenomenon is not determinist

29、ic. Random process. Stochastic process is the probability model based on probability space, is considered the dynamics of probability theory, namely, it is the study of the random phenomena with time evolution. So the random behavior is a kind of statistical rules of behavior. The price behavior mod

30、els are usually expressed in the famous Wiener process. We assume that the Wiener process the stock price following the general is tempting, that is to say, it has the expected drift rate and variance rate. The Wiener process indicates that only the values of variables associated with future project

31、ions, past history variables and variables from the past to the present and the future evolution of the way the stock price forecast is not relevant. The Maldives Kopf and the nature of the weak market efficiency (the weak form of market efficiency) is consistent, that is to say, a stock price alrea

32、dy contains all the information, including all the past records of prices. But when people began to study the financial market by using fractal theory, found its operation does not follow the Brown campaign but to more general fractional Brown motion.Brown movement throughout the financial field, in

33、 the modern financial field occupies an important position.Brown refers to a random walk without correlation, meet the statistical self similarity characteristics with random fractal, but the function of time (trajectories) self affine function is available. Said: (as) (0) Xtt is the standard Brown

34、motion, then each of the following is the standard Brown random function 1, (21).) (/) from XtcXtc (c > 0 is a constant, t = 0), 2 (2) () (XtXthhX) (H > 0 is a constant, t = 0), 13 (3) (0) (0) (0) tXttXttThe motion of particles is very much in the form of translation and its transfer. It is ob

35、vious that the particles are not related to each other. Even when the particles are close to each other, the particle motion is not affected by the particle size.Bach Leigh Ye in France in 1900, the speculation theory, the stock price is also seen as a random movement, the resulting equation is very

36、 similar to the description of the Brown particle motion, but the resulting stock price may be negative, apparently inconsistent with the actual, Bach Leigh Ye so-called "Brown movement", in other words, he put the stock price changes, the ideal for the Brown movement before the study, it

37、represents the meaning of "no rules", as long as the economy The research of control theory. Absorbed by founder Weiner in 1923 on the Brown campaign made a strict mathematical definition, according to this definition, the Brown movement is a kind of independent increment process, so it is

38、 a Markov process, the mathematical community is often the Brown campaign called Weiner.Markowiz (1952): the theory of portfolio selection; Roberts Osborne (1959) and the stock market research into the concept of random walk and the number of the Brown movement; Samuelson and Fama (1970) of the effe

39、ctive market theory (EMH); Fischer Black And (Black-Scholes model); Ross (1976) of the arbitrage pricing theory.In this century, after more than half a century of research, people gradually get close to it. In the beginning of this century, the theory of Smoluchowski and Einstein, and Berg Si Weide&

40、#39;s experiments, made this important scientific problem solving. This is the real existence of molecules. This is a direct and convincing evidence, which is the end of the debate on the nature of atoms and molecules. "The Brown movement and the dynamic hypothesis are consistent, and this make

41、s the most discerning scientist have to admit that this is a matter of space," said Peng Jiale, a mathematician and physicist, who concluded in 1913 that "the atomic number of atoms is a real" and that the theory is applied in other fields as well as the background noise in the circuit. Study on the Brown movement and the thermal motion of the molecules are not the s