布朗运动及其应用

随机过程在金融领域的作用随机过程在金融领域的作用 1240410114 王颖 浅谈布朗运动在金融领域的应用浅谈布朗运动在金融领域的应用 悬浮微粒永不停息地做无规则运动的现象叫做布朗运动 例如 在显微镜下观察悬浮在水中的藤黄粉 花粉微粒 或在无风情形观察 空气中的烟粒 尘埃时都会看到这种运动 温度越高 运动越激烈 它是 1827 年植物学家 R 布朗首先发现的 作布朗运动的粒子非常微小 直径约 1 10 微米 在周围液体或气体分子的碰撞下 产生一种涨落不定的净作用 力 导致微粒的布朗运动 如果布朗粒子相互碰撞的机会很少 可以看成是 巨大分子组成的理想气体 则在重力场中达到热平衡后 其数密度按高度的 分布应遵循玻耳兹曼分布 J B 佩兰的实验证实了这一点 并由此相当精 确地测定了阿伏伽德罗常量 及一系列与微粒有关的数据 1905 年 A 爱因斯 坦根据扩散方程建立了布朗运动的统计理论 布朗运动的发现 实验研究和 理论分析间接地证实了分子的无规则 热运动 对于气体动理论 的建立以及 确认物质结构的原子性具有重要意义 并且推动统计物理学特别是涨落理论 的发展 由于布朗运动代表一种随机涨落现象 它的理论对于仪表测量精度 限制的研究以及高倍放大电讯电路中背景噪声的研究等有广泛应用 这 是 1826 年英国植物学家布朗 1773 1858 用显微镜观察悬浮在水中的花 粉时发现的 后来把悬浮微粒的这种运动叫做布朗运动 不只是花粉和小炭 粒 对于液体中各种不同的悬浮微粒 都可以观察到布朗运动 布朗的发现是一个新奇的现象 它的原因是什么 人们是迷惑不解的 在布 朗之后 这一问题一再被提出 为此有许多学者进行过长期的研究 一些早 期的研究者简单地把它归结为热或电等外界因素引起的 最早隐约指向合理 解释的是维纳 1826 1896 1863 年他提出布朗运动起源于分子的振动 他还公布了首次对微粒速度与粒度关系的观察结果 不过他的分子模型还不 是现代的模型 他看到的实际上是微粒的位移 并不是振动 到了 70 80 年代 一些学者明确地把布朗运动归结为液体分子撞击微粒 的结果 这些学者有卡蓬内尔 德尔索和梯瑞昂 还有耐格里 植物学家耐 格里 1879 从真菌 细菌等通过空气传播的现象 认为这些微粒即使在静止 的空气中也可以不沉 联系到物理学中气体分子以很高速度向各方向运动的 结论 他推测在阳光下看到的飞舞的尘埃是气体分子从各方向撞击的结果 他说 这些微小尘埃就象弹性球一样被掷来掷去 结果如同分子本身一样 能保持长久的悬浮 不过耐格里又放弃了这一可能达到正确解释的途径 他计算了单个气体分子和尘埃微粒发生弹性碰撞时微粒的速度 结果要比实 际观察到的小许多数量级 于是他认为由于气体分子运动的无规则性 它们 共同作用的结果不能使微粒达到观察速度值 而在液体中则由于介质和微粒 的摩擦阻力和分子间的粘附力 分子运动的设想不能成为合适的解释 无规则行走 random walk 定义 无规则行走就是随机游走 其概念接近于布朗运动 是是布布朗朗运运 动动的的理理想想数数学学状状态态 核心概念 任任何何无无规规则则行行走走者者所所带带的的守守恒恒量量都都各各自自对对应应着着一一个个扩扩散散运运输输 定定律律 就这样 布朗运动自发现之后 经过多半个世纪的研究 人们逐渐接近 对它的正确认识 到本世纪初 先是爱因斯坦和斯莫卢霍夫斯基的理论 然 后是贝兰和斯维德伯格的实验使这一重大的科学问题得到圆满地解决 并首 次测定了阿伏加德罗常数 这也就是为分子的真实存在提供了一个直观的 令人信服的证据 这对基础科学和哲学有着巨大的意义 从这以后 科学上 关于原子和分子真实性的争论即告终结 正如原先原子论的主要反对者奥斯 特瓦尔德所说 布朗运动和动力学假说的一致 已经被贝兰十分圆满地证 实了 这就使那怕最挑剔的科学家也得承认这是充满空间的物质的原子构成 的一个实验证据 数学家和物理学家彭加勒在 1913 年总结性地说道 贝兰对原子数目的光辉测定完成了原子论的胜利 化学家的原子论 现在是一个真实存在 布朗运动代表了一种随机涨落现象 它的理论在其 他领域也有重要应用 如对测量仪器的精度限度的研究 高倍放大电讯电路 中的背景噪声的研究等 布朗运动与分子热运动不一样 与温度和粒子个数 有关 温度越高 布朗运动越剧烈 粒子越少 分子热运动越剧烈 分子 永不停息地做无规则的运动 分子永不停息地做无规则的运动 布朗运动 扩散现象都说明了任何物质的分子 不论在什么状态下 都在做永不停息的 无规则运动 分子的无规则运动与物质的温度有关 温度越高 分子的无规 则运动越剧烈 将布朗运动与股票价格行为联系在一起 进而建立起维纳过程的数学模 型是本世纪的一项具有重要意义的金融创新 在现代金融数学中占有重要地 位 迄今 普遍的观点仍认为 股票市场是随机波动的 随机波动是股票市 场最根本的特性 是股票市场的常态 布朗运动假设是现代资本市场理论的核心假设 现代资本市场理论认为 证券期货价格具有随机性特征 这里的所谓随机性 是指数据的无记忆性 即过去数据不构成对未来数据的预测基础 同时不会出现惊人相似的反复 随机现象的数学定义是 在个别试验中其结果呈现出不确定性 在大量重复 试验中其结果又具有统计规律性的现象 描述股价行为模型之一的布朗运动 之维纳过程是马尔科夫随机过程的一种特殊形式 而马尔科夫过程是一种特 殊类型的随机过程 随机过程是建立在概率空间上的概率模型 被认为是概 率论的动力学 即它的研究对象是随时间演变的随机现象 所以随机行为是 一种具有统计规律性的行为 股价行为模型通常用著名的维纳过程来表达 假定股票价格遵循一般化的维纳过程是很具诱惑力的 也就是说 它具有不 变的期望漂移率和方差率 维纳过程说明只有变量的当前值与未来的预测有 关 变量过去的历史和变量从过去到现在的演变方式则与未来的预测不相关 股价的马尔科夫性质与弱型市场有效性 the weak form of market efficiency 相一致 也就是说 一种股票的现价已经包含了所有信息 当 然包括了所有过去的价格记录 但是当人们开始采用分形理论研究金融市场 时 发现它的运行并不遵循布朗运动 而是服从更为一般的分数布朗运动 布朗运动贯穿于金融领域中 在现代金融领域中占有重要地位 布朗运动指的是一种无相关性的随机行走 满足统计自相似性 即具有 随机分形的特征 但其时间函数 运动轨迹 却是自仿射的 可用函数表示 如 0 Xtt 是标准布朗运动 则下列各个随机函数也是标准布朗运动 1 21 XtcXtc c 0 为常数 t 0 2 2 XtXthhX h 0 为常数 t 0 3 13 0 0 0 tXttXtt 具有以下主要特性 粒子的运动由平移及其转移所构成 显得非常没规 则而且其轨迹几乎是处处没有切线 粒子之移动显然互不相关 甚至于当粒 子互相接近至比其直径小的距离时也是如此 粒子越小或液体粘性越低或温 度越高时 粒子的运动越活泼 粒子的成分及密度对其运动没有影响 粒子 的运动永不停止 二 布朗运动在金融领域的发展 1900 年法国的巴施利叶在博士论文 投机理论 中将股票价格的涨 跌也看作是一种随机运动 所得到的方程与描述布朗粒子运动的方程非常相 似 第一次给予布朗运动以严格的数学描述 但由此得到的股票价格可能取 负值 显然与实际不符当然 巴施利叶所谓的 布朗运动 实质上指的 是股市的价格变动 换句话说 他把股价的变动 理想化为布朗运动 可见 在物理学界尚未把布朗运动研究清楚之前 它象征 无规行走 的意义 早 就被经济研究所吸纳了 控制论创始人维纳于1923 年对布朗运动作出了严 格的数学定义 根据这一定义 布朗运动是一种独立增量过程 因而是一种 马尔科夫过程 数学界也常把布朗运动称为维纳过程 Markowiz 1952 发 表投资组合选择理论 Roberts 和 Osborne 1959 把随机数游走和布朗运动 的概念带入股市研究 Samuelson 和 Fama 1970 的有效市场理论 EMH Fischer Black 和 Black Scholes 模型 Ross 1976 的套利定价理 布朗运动假设是现代资本市场理论的核心假设 现代资本市场理论认为 证券期货价格具有随机性特征 这里的所谓随机性 是指数据的无记忆性 即过去数据不构成对未来数据的预测基础 同时不会出现惊人相似的反复 股价行为模型通常用著名的维纳过程来表达 假定股票价格遵循一般化的维 纳过程是很具诱惑力的 也就是说 它具有不变的期望漂移率和方差率 将布朗运动与股票价格行为联系在一起 进而建立起维纳过程的数学模型 是本世纪的一项具有重要意义的金融创新 在现代金融数学中占有重要地位 迄今 普遍的观点仍认为 股票市场是随机波动的 随机波动是股票市场最根本的特性 是股票市场的常态 布朗运动假设 是现代资本市场理论的核心假设 现代资本市场理论认为证券期货价格具有 随机性特征 这里的所谓随机性 是指数据的无记忆性 即过去数据不构成 对未来数据的预测基础 同时不会出现惊人相似的反复 英英文文翻翻译译 On the application of Brown in the financial field The phenomenon of suspended particles never cease to do no regular movement is called 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 discovered Brown motion of the particle diameter is very small 1 10 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 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 particle 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 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 telecommunications 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 different 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 external 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 models 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 spread 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 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 actual 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 motion 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 corresponding 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 core 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 deterministic 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 models 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 projections 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 already 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 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 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 tXttXtt The motion of particles is very much in the form of translation and its transfer It is obvious 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 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 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 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 effective 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 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 makes 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 n