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db 3.json
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文章分类\ndate: 2019-01-14 20:53:04\ntype: \"categories\" #这部分是新添加的\n---\n","updated":"2022-01-22T07:42:49.631Z","path":"categories/index.html","comments":1,"layout":"page","_id":"ckypkquxr00003opk5mvz11xm","content":"","site":{"data":{}},"excerpt":"","more":""},{"title":"标签","date":"2019-01-14T12:56:48.000Z","type":"tags","_content":"","source":"tags/index.md","raw":"---\ntitle: 标签\ndate: 2019-01-14 20:56:48\ntype: \"tags\" #新添加的内容\n---\n","updated":"2022-01-22T07:42:36.591Z","path":"tags/index.html","comments":1,"layout":"page","_id":"ckypkquxz00023opk9yps5nh6","content":"","site":{"data":{}},"excerpt":"","more":""}],"Post":[{"title":"hello world!","_content":"\n这是我的第1篇post","source":"_posts/hello world.md","raw":"---\ntitle: hello world!\n---\n\n这是我的第1篇post","slug":"hello world","published":1,"date":"2022-01-22T07:38:13.692Z","updated":"2022-01-22T07:47:04.314Z","comments":1,"layout":"post","photos":[],"link":"","_id":"ckypkquxv00013opk3wtzekoo","content":"<p>这是我的第1篇post</p>\n","site":{"data":{}},"excerpt":"","more":"<p>这是我的第1篇post</p>\n"},{"layout":"posts","title":"Stochastic Calculus for Finance","_content":"\n<head>\n <script src=\"https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\" type=\"text/javascript\"></script>\n <script type=\"text/x-mathjax-config\">\n MathJax.Hub.Config({\n tex2jax: {\n skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'],\n inlineMath: [['$','$']]\n }\n });\n </script>\n</head>\n\n\n# General Probability Theory\n\n## infinite probability spaces\n\n### $\\sigma$-代数\n\n$\\sigma-algebra$, F is a collection of subsets of $\\Omega$\n\n1. empty set belongs to it\n2. whenever a set belongs to it, its complement also belong to F\n3. the union of a sequence of sets in F belongs to F\n\n可以有推论：\n\n1. $\\Omega$ 一定在F中\n2. 集合序列的交集也一定在F中\n\n### 概率测量的定义 probability measure\n\n一个函数，定义域是F中的任意集合，值域是[0,1]，满足：\n\n1. $P(\\Omega)=1$\n2. $P(\\cup^\\infty_{n=1} A_n)=\\sum^\\infty_{n=1} P(A_n)$\n \n $triple(\\Omega,F,P)$称为概率空间\n \n\nuniform（Lebesgue）measure on [0,1]：在0，1之间选取一个数，定义：\n\n$P(a,b)=P[a,b]=b-a,0\\le a \\le b\\le 1$\n\n可以用上述方式表述概率的集合的集合构成了sigma代数，从包含的所有闭区间出发，称为Borel sigma代数\n\n$(a,b)=\\cup_{n=1}^\\infty[a+\\frac{1}{n},b-\\frac{1}{n}]$ ，所以sigma代数中包含了所有开区间，分别取补集可以导出包含开区间与闭区间的并，从而导出所有集合 \n\n通过从闭区间出发，添加所有必要元素构成的sigma代数称之为$Borel \\quad\\sigma-algebra$ of subsets of [0,1] and is denoted B[0,1]\n\nevent A occurs almost surely if P(A)=1\n\n## Random variables and distributions\n\ndefinition: a random variable is a real-valued function X defined on $\\Omega$ with the property that for every Borel subset B of **R,** the subset of $\\Omega$ given by:\n\n$$\n\\begin{equation*}\\{x\\in B\\}=\\{\\omega \\in \\Omega ; X(\\omega )\\in B\\}\\end{equation*}\n$$\n\nis in the $\\sigma$-algebra F\n\n本质是把事件映射为实数，同时为了保证可以拟映射，要求函数可测。\n\n构造R的Borel子集？从所有的闭区间出发，闭区间的交——特别地，开区间也包含进来，从而开集包含进来 ，因为每个开集可以写成开区间序列的并。闭集也是Borel集合，因为是开集的补集。\n\n关注X取值包含于某集合而不是具体的值\n\nDefinition: let X be a random variable on a probability space, the distribution measure of X is the probability measure $\\mu _X$that assigns to each Borel subset B of R the mass $\\mu _X(B)=P\\{X\\in B\\}$ \n\nRandom variable 有distribution 但两者不等同，两个不同的Random variable 可以有相同的distribution，一个单独的random variable 可以有两个不同的distribution\n\ncdf：$F(x)=P\\{X\\le x\\}$\n\n$\\mu_X(x,y]=F(y)-F(x)$\n\n## Expectations\n\n$E(X)=\\sum X(\\omega )P(\\omega )$\n\n$\\{\\Omega , F, P\\}$ random variable $X(\\omega )$ P是概率空间中的测度\n\n$A_k=\\{\\omega \\in \\Omega ; y_k\\le X(\\omega ) < y_{k+1}\\}$\n\nlower Lebesgue sum $LS^-_{\\Pi}=\\sum y_k P(A_k)$\n\nthe maximal distance between the $y_k$ partition points approaches zero, we get Lebesgue integral$\\int_{\\Omega}X(\\omega )dP(\\omega)$\n\nLebesgue integral相当于把积分概念拓展到可测空间，而不是单纯的更换了求和方式。横坐标实际上是$\\Omega$的测度。\n\nif the random variables X can take both positive and negative values, we can define the positive and negative parts of X:\n\n$X^+=max\\{X,0\\}$, $X^-=min\\{-X,0\\}$\n\n$\\int XdP=\\int X^+dP-\\int X^- dP$\n\n**Comparison**\n\nIf $X\\le Y$ almost surely, and if the Lebesgue integral are defined, then\n\n$\\int _{\\Omega}X(\\omega)dP(\\omega)\\le \\int _{\\Omega}Y(\\omega)dP(\\omega)$\n\nIf $X=Y$ almost surely, and if the Lebesgue integral are defined, then\n\n$\\int _{\\Omega}X(\\omega)dP(\\omega)=\\int _{\\Omega}Y(\\omega)dP(\\omega)$\n\n**Integrability, Linearity ...**\n\n**Jensen’s inequality**\n\nif $\\phi$ is a convex, real-valued function defined on R and if $E|X|<\\infty$ ,then\n\n$\\phi(EX)\\le E\\phi(X)$\n\n$\\mathcal{B} (\\mathbb{R})$ be the sigma-algebra of Borel subsets of $\\mathbb{R}$, the Lebesgue measure $\\mathcal{L}$ on R assigns to each set $B\\in \\mathcal{B}(\\mathbb{R})$ a number in $[0,\\infty)$ or the value $\\infty$ so that:\n\n1. $\\mathcal{L}[a,b] =b-a$ \n2. if $B_1,B_2,B_3 \\dots$ is a sequence of disjoint sets in $\\mathcal{B}$ ,then we have the countable additivity property: $\\mathcal{L} (\\cup_{n=1}^\\infty B_n)=\\sum_{n=1}^\\infty \\mathcal{L} (B_n)$\n\n黎曼积分有且只有在区间中非连续点的集合的Lebesgue测度为零时有定义，即f在区间上几乎处处连续\n\n若f的Riemann积分在区间上存在，则f是Borel可测的，而且Riemann积分和Lebesgue积分一致。\n\n### convergence of integrals\n\n**definition**\n\n$X_1,X_2,X_3\\dots$ be a sequence of random variables on the same probability space$(\\Omega, \\mathcal{F},\\mathbb{P})$, $X$ be another random variable. $X_1,X_2,X_3\\dots$ converges to $X$ almost surely $lim_{n\\to \\infty}X_n=X$ almost surely. if the set of $\\omega \\in \\Omega$ for the sequence has limit $X(\\omega)$ is a set with probability one. \n\nStrong law of Large Numbers: \n\n实的Borel可测的函数列$f_1,f_2,f_3\\dots$ defined on$\\mathbb{R}$, $f$ 也是实的Borel可测函数，the sequence converges to f almost every-where if 序列极限不为f的点的集合的 Lebesgue measure为零\n\n$lim_{n\\to \\infty}f_n=f \\textit{ almost everywhere}$","source":"_posts/Stochastic Calculus.md","raw":"---\nlayout: posts\ntitle: Stochastic Calculus for Finance\ncategories: 学习笔记\ntags: [金融,数学]\n\n---\n\n<head>\n <script src=\"https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\" type=\"text/javascript\"></script>\n <script type=\"text/x-mathjax-config\">\n MathJax.Hub.Config({\n tex2jax: {\n skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'],\n inlineMath: [['$','$']]\n }\n });\n </script>\n</head>\n\n\n# General Probability Theory\n\n## infinite probability spaces\n\n### $\\sigma$-代数\n\n$\\sigma-algebra$, F is a collection of subsets of $\\Omega$\n\n1. empty set belongs to it\n2. whenever a set belongs to it, its complement also belong to F\n3. the union of a sequence of sets in F belongs to F\n\n可以有推论：\n\n1. $\\Omega$ 一定在F中\n2. 集合序列的交集也一定在F中\n\n### 概率测量的定义 probability measure\n\n一个函数，定义域是F中的任意集合，值域是[0,1]，满足：\n\n1. $P(\\Omega)=1$\n2. $P(\\cup^\\infty_{n=1} A_n)=\\sum^\\infty_{n=1} P(A_n)$\n \n $triple(\\Omega,F,P)$称为概率空间\n \n\nuniform（Lebesgue）measure on [0,1]：在0，1之间选取一个数，定义：\n\n$P(a,b)=P[a,b]=b-a,0\\le a \\le b\\le 1$\n\n可以用上述方式表述概率的集合的集合构成了sigma代数，从包含的所有闭区间出发，称为Borel sigma代数\n\n$(a,b)=\\cup_{n=1}^\\infty[a+\\frac{1}{n},b-\\frac{1}{n}]$ ，所以sigma代数中包含了所有开区间，分别取补集可以导出包含开区间与闭区间的并，从而导出所有集合 \n\n通过从闭区间出发，添加所有必要元素构成的sigma代数称之为$Borel \\quad\\sigma-algebra$ of subsets of [0,1] and is denoted B[0,1]\n\nevent A occurs almost surely if P(A)=1\n\n## Random variables and distributions\n\ndefinition: a random variable is a real-valued function X defined on $\\Omega$ with the property that for every Borel subset B of **R,** the subset of $\\Omega$ given by:\n\n$$\n\\begin{equation*}\\{x\\in B\\}=\\{\\omega \\in \\Omega ; X(\\omega )\\in B\\}\\end{equation*}\n$$\n\nis in the $\\sigma$-algebra F\n\n本质是把事件映射为实数，同时为了保证可以拟映射，要求函数可测。\n\n构造R的Borel子集？从所有的闭区间出发，闭区间的交——特别地，开区间也包含进来，从而开集包含进来 ，因为每个开集可以写成开区间序列的并。闭集也是Borel集合，因为是开集的补集。\n\n关注X取值包含于某集合而不是具体的值\n\nDefinition: let X be a random variable on a probability space, the distribution measure of X is the probability measure $\\mu _X$that assigns to each Borel subset B of R the mass $\\mu _X(B)=P\\{X\\in B\\}$ \n\nRandom variable 有distribution 但两者不等同，两个不同的Random variable 可以有相同的distribution，一个单独的random variable 可以有两个不同的distribution\n\ncdf：$F(x)=P\\{X\\le x\\}$\n\n$\\mu_X(x,y]=F(y)-F(x)$\n\n## Expectations\n\n$E(X)=\\sum X(\\omega )P(\\omega )$\n\n$\\{\\Omega , F, P\\}$ random variable $X(\\omega )$ P是概率空间中的测度\n\n$A_k=\\{\\omega \\in \\Omega ; y_k\\le X(\\omega ) < y_{k+1}\\}$\n\nlower Lebesgue sum $LS^-_{\\Pi}=\\sum y_k P(A_k)$\n\nthe maximal distance between the $y_k$ partition points approaches zero, we get Lebesgue integral$\\int_{\\Omega}X(\\omega )dP(\\omega)$\n\nLebesgue integral相当于把积分概念拓展到可测空间，而不是单纯的更换了求和方式。横坐标实际上是$\\Omega$的测度。\n\nif the random variables X can take both positive and negative values, we can define the positive and negative parts of X:\n\n$X^+=max\\{X,0\\}$, $X^-=min\\{-X,0\\}$\n\n$\\int XdP=\\int X^+dP-\\int X^- dP$\n\n**Comparison**\n\nIf $X\\le Y$ almost surely, and if the Lebesgue integral are defined, then\n\n$\\int _{\\Omega}X(\\omega)dP(\\omega)\\le \\int _{\\Omega}Y(\\omega)dP(\\omega)$\n\nIf $X=Y$ almost surely, and if the Lebesgue integral are defined, then\n\n$\\int _{\\Omega}X(\\omega)dP(\\omega)=\\int _{\\Omega}Y(\\omega)dP(\\omega)$\n\n**Integrability, Linearity ...**\n\n**Jensen’s inequality**\n\nif $\\phi$ is a convex, real-valued function defined on R and if $E|X|<\\infty$ ,then\n\n$\\phi(EX)\\le E\\phi(X)$\n\n$\\mathcal{B} (\\mathbb{R})$ be the sigma-algebra of Borel subsets of $\\mathbb{R}$, the Lebesgue measure $\\mathcal{L}$ on R assigns to each set $B\\in \\mathcal{B}(\\mathbb{R})$ a number in $[0,\\infty)$ or the value $\\infty$ so that:\n\n1. $\\mathcal{L}[a,b] =b-a$ \n2. if $B_1,B_2,B_3 \\dots$ is a sequence of disjoint sets in $\\mathcal{B}$ ,then we have the countable additivity property: $\\mathcal{L} (\\cup_{n=1}^\\infty B_n)=\\sum_{n=1}^\\infty \\mathcal{L} (B_n)$\n\n黎曼积分有且只有在区间中非连续点的集合的Lebesgue测度为零时有定义，即f在区间上几乎处处连续\n\n若f的Riemann积分在区间上存在，则f是Borel可测的，而且Riemann积分和Lebesgue积分一致。\n\n### convergence of integrals\n\n**definition**\n\n$X_1,X_2,X_3\\dots$ be a sequence of random variables on the same probability space$(\\Omega, \\mathcal{F},\\mathbb{P})$, $X$ be another random variable. $X_1,X_2,X_3\\dots$ converges to $X$ almost surely $lim_{n\\to \\infty}X_n=X$ almost surely. if the set of $\\omega \\in \\Omega$ for the sequence has limit $X(\\omega)$ is a set with probability one. \n\nStrong law of Large Numbers: \n\n实的Borel可测的函数列$f_1,f_2,f_3\\dots$ defined on$\\mathbb{R}$, $f$ 也是实的Borel可测函数，the sequence converges to f almost every-where if 序列极限不为f的点的集合的 Lebesgue measure为零\n\n$lim_{n\\to \\infty}f_n=f \\textit{ almost everywhere}$","slug":"Stochastic Calculus","published":1,"date":"2022-01-22T07:46:20.584Z","updated":"2022-01-22T08:31:01.009Z","comments":1,"photos":[],"link":"","_id":"ckypkquy000033opk9jpicgn7","content":"<head>\n <script src=\"https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\" type=\"text/javascript\"></script>\n <script type=\"text/x-mathjax-config\">\n MathJax.Hub.Config({\n tex2jax: {\n skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'],\n inlineMath: [['$','$']]\n }\n });\n </script>\n</head>\n\n\n<h1 id=\"General-Probability-Theory\"><a href=\"#General-Probability-Theory\" class=\"headerlink\" title=\"General Probability Theory\"></a>General Probability Theory</h1><h2 id=\"infinite-probability-spaces\"><a href=\"#infinite-probability-spaces\" class=\"headerlink\" title=\"infinite probability spaces\"></a>infinite probability spaces</h2><h3 id=\"sigma-代数\"><a href=\"#sigma-代数\" class=\"headerlink\" title=\"$\\sigma$-代数\"></a>$\\sigma$-代数</h3><p>$\\sigma-algebra$, F is a collection of subsets of $\\Omega$</p>\n<ol>\n<li>empty set belongs to it</li>\n<li>whenever a set belongs to it, its complement also belong to F</li>\n<li>the union of a sequence of sets in F belongs to F</li>\n</ol>\n<p>可以有推论：</p>\n<ol>\n<li>$\\Omega$ 一定在F中</li>\n<li>集合序列的交集也一定在F中</li>\n</ol>\n<h3 id=\"概率测量的定义-probability-measure\"><a href=\"#概率测量的定义-probability-measure\" class=\"headerlink\" title=\"概率测量的定义 probability measure\"></a>概率测量的定义 probability measure</h3><p>一个函数，定义域是F中的任意集合，值域是[0,1]，满足：</p>\n<ol>\n<li>$P(\\Omega)=1$</li>\n<li>$P(\\cup^\\infty_{n=1} A_n)=\\sum^\\infty_{n=1} P(A_n)$ $triple(\\Omega,F,P)$称为概率空间</li>\n</ol>\n<p>uniform（Lebesgue）measure on [0,1]：在0，1之间选取一个数，定义：</p>\n<p>$P(a,b)=P[a,b]=b-a,0\\le a \\le b\\le 1$</p>\n<p>可以用上述方式表述概率的集合的集合构成了sigma代数，从包含的所有闭区间出发，称为Borel sigma代数</p>\n<p>$(a,b)=\\cup_{n=1}^\\infty[a+\\frac{1}{n},b-\\frac{1}{n}]$ ，所以sigma代数中包含了所有开区间，分别取补集可以导出包含开区间与闭区间的并，从而导出所有集合 </p>\n<p>通过从闭区间出发，添加所有必要元素构成的sigma代数称之为$Borel \\quad\\sigma-algebra$ of subsets of [0,1] and is denoted B[0,1]</p>\n<p>event A occurs almost surely if P(A)=1</p>\n<h2 id=\"Random-variables-and-distributions\"><a href=\"#Random-variables-and-distributions\" class=\"headerlink\" title=\"Random variables and distributions\"></a>Random variables and distributions</h2><p>definition: a random variable is a real-valued function X defined on $\\Omega$ with the property that for every Borel subset B of <strong>R,</strong> the subset of $\\Omega$ given by:</p>\n<p>$$<br>\\begin{equation*}{x\\in B}={\\omega \\in \\Omega ; X(\\omega )\\in B}\\end{equation*}<br>$$</p>\n<p>is in the $\\sigma$-algebra F</p>\n<p>本质是把事件映射为实数，同时为了保证可以拟映射，要求函数可测。</p>\n<p>构造R的Borel子集？从所有的闭区间出发，闭区间的交——特别地，开区间也包含进来，从而开集包含进来 ，因为每个开集可以写成开区间序列的并。闭集也是Borel集合，因为是开集的补集。</p>\n<p>关注X取值包含于某集合而不是具体的值</p>\n<p>Definition: let X be a random variable on a probability space, the distribution measure of X is the probability measure $\\mu _X$that assigns to each Borel subset B of R the mass $\\mu _X(B)=P{X\\in B}$ </p>\n<p>Random variable 有distribution 但两者不等同，两个不同的Random variable 可以有相同的distribution，一个单独的random variable 可以有两个不同的distribution</p>\n<p>cdf：$F(x)=P{X\\le x}$</p>\n<p>$\\mu_X(x,y]=F(y)-F(x)$</p>\n<h2 id=\"Expectations\"><a href=\"#Expectations\" class=\"headerlink\" title=\"Expectations\"></a>Expectations</h2><p>$E(X)=\\sum X(\\omega )P(\\omega )$</p>\n<p>${\\Omega , F, P}$ random variable $X(\\omega )$ P是概率空间中的测度</p>\n<p>$A_k={\\omega \\in \\Omega ; y_k\\le X(\\omega ) < y_{k+1}}$</p>\n<p>lower Lebesgue sum $LS^-_{\\Pi}=\\sum y_k P(A_k)$</p>\n<p>the maximal distance between the $y_k$ partition points approaches zero, we get Lebesgue integral$\\int_{\\Omega}X(\\omega )dP(\\omega)$</p>\n<p>Lebesgue integral相当于把积分概念拓展到可测空间，而不是单纯的更换了求和方式。横坐标实际上是$\\Omega$的测度。</p>\n<p>if the random variables X can take both positive and negative values, we can define the positive and negative parts of X:</p>\n<p>$X^+=max{X,0}$, $X^-=min{-X,0}$</p>\n<p>$\\int XdP=\\int X^+dP-\\int X^- dP$</p>\n<p><strong>Comparison</strong></p>\n<p>If $X\\le Y$ almost surely, and if the Lebesgue integral are defined, then</p>\n<p>$\\int _{\\Omega}X(\\omega)dP(\\omega)\\le \\int _{\\Omega}Y(\\omega)dP(\\omega)$</p>\n<p>If $X=Y$ almost surely, and if the Lebesgue integral are defined, then</p>\n<p>$\\int _{\\Omega}X(\\omega)dP(\\omega)=\\int _{\\Omega}Y(\\omega)dP(\\omega)$</p>\n<p><strong>Integrability, Linearity …</strong></p>\n<p><strong>Jensen’s inequality</strong></p>\n<p>if $\\phi$ is a convex, real-valued function defined on R and if $E|X|<\\infty$ ,then</p>\n<p>$\\phi(EX)\\le E\\phi(X)$</p>\n<p>$\\mathcal{B} (\\mathbb{R})$ be the sigma-algebra of Borel subsets of $\\mathbb{R}$, the Lebesgue measure $\\mathcal{L}$ on R assigns to each set $B\\in \\mathcal{B}(\\mathbb{R})$ a number in $[0,\\infty)$ or the value $\\infty$ so that:</p>\n<ol>\n<li>$\\mathcal{L}[a,b] =b-a$ </li>\n<li>if $B_1,B_2,B_3 \\dots$ is a sequence of disjoint sets in $\\mathcal{B}$ ,then we have the countable additivity property: $\\mathcal{L} (\\cup_{n=1}^\\infty B_n)=\\sum_{n=1}^\\infty \\mathcal{L} (B_n)$</li>\n</ol>\n<p>黎曼积分有且只有在区间中非连续点的集合的Lebesgue测度为零时有定义，即f在区间上几乎处处连续</p>\n<p>若f的Riemann积分在区间上存在，则f是Borel可测的，而且Riemann积分和Lebesgue积分一致。</p>\n<h3 id=\"convergence-of-integrals\"><a href=\"#convergence-of-integrals\" class=\"headerlink\" title=\"convergence of integrals\"></a>convergence of integrals</h3><p><strong>definition</strong></p>\n<p>$X_1,X_2,X_3\\dots$ be a sequence of random variables on the same probability space$(\\Omega, \\mathcal{F},\\mathbb{P})$, $X$ be another random variable. $X_1,X_2,X_3\\dots$ converges to $X$ almost surely $lim_{n\\to \\infty}X_n=X$ almost surely. if the set of $\\omega \\in \\Omega$ for the sequence has limit $X(\\omega)$ is a set with probability one. </p>\n<p>Strong law of Large Numbers: </p>\n<p>实的Borel可测的函数列$f_1,f_2,f_3\\dots$ defined on$\\mathbb{R}$, $f$ 也是实的Borel可测函数，the sequence converges to f almost every-where if 序列极限不为f的点的集合的 Lebesgue measure为零</p>\n<p>$lim_{n\\to \\infty}f_n=f \\textit{ almost everywhere}$</p>\n","site":{"data":{}},"excerpt":"","more":"<head>\n <script src=\"https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\" type=\"text/javascript\"></script>\n <script type=\"text/x-mathjax-config\">\n MathJax.Hub.Config({\n tex2jax: {\n skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'],\n inlineMath: [['$','$']]\n }\n });\n </script>\n</head>\n\n\n<h1 id=\"General-Probability-Theory\"><a href=\"#General-Probability-Theory\" class=\"headerlink\" title=\"General Probability Theory\"></a>General Probability Theory</h1><h2 id=\"infinite-probability-spaces\"><a href=\"#infinite-probability-spaces\" class=\"headerlink\" title=\"infinite probability spaces\"></a>infinite probability spaces</h2><h3 id=\"sigma-代数\"><a href=\"#sigma-代数\" class=\"headerlink\" title=\"$\\sigma$-代数\"></a>$\\sigma$-代数</h3><p>$\\sigma-algebra$, F is a collection of subsets of $\\Omega$</p>\n<ol>\n<li>empty set belongs to it</li>\n<li>whenever a set belongs to it, its complement also belong to F</li>\n<li>the union of a sequence of sets in F belongs to F</li>\n</ol>\n<p>可以有推论：</p>\n<ol>\n<li>$\\Omega$ 一定在F中</li>\n<li>集合序列的交集也一定在F中</li>\n</ol>\n<h3 id=\"概率测量的定义-probability-measure\"><a href=\"#概率测量的定义-probability-measure\" class=\"headerlink\" title=\"概率测量的定义 probability measure\"></a>概率测量的定义 probability measure</h3><p>一个函数，定义域是F中的任意集合，值域是[0,1]，满足：</p>\n<ol>\n<li>$P(\\Omega)=1$</li>\n<li>$P(\\cup^\\infty_{n=1} A_n)=\\sum^\\infty_{n=1} P(A_n)$ $triple(\\Omega,F,P)$称为概率空间</li>\n</ol>\n<p>uniform（Lebesgue）measure on [0,1]：在0，1之间选取一个数，定义：</p>\n<p>$P(a,b)=P[a,b]=b-a,0\\le a \\le b\\le 1$</p>\n<p>可以用上述方式表述概率的集合的集合构成了sigma代数，从包含的所有闭区间出发，称为Borel sigma代数</p>\n<p>$(a,b)=\\cup_{n=1}^\\infty[a+\\frac{1}{n},b-\\frac{1}{n}]$ ，所以sigma代数中包含了所有开区间，分别取补集可以导出包含开区间与闭区间的并，从而导出所有集合 </p>\n<p>通过从闭区间出发，添加所有必要元素构成的sigma代数称之为$Borel \\quad\\sigma-algebra$ of subsets of [0,1] and is denoted B[0,1]</p>\n<p>event A occurs almost surely if P(A)=1</p>\n<h2 id=\"Random-variables-and-distributions\"><a href=\"#Random-variables-and-distributions\" class=\"headerlink\" title=\"Random variables and distributions\"></a>Random variables and distributions</h2><p>definition: a random variable is a real-valued function X defined on $\\Omega$ with the property that for every Borel subset B of <strong>R,</strong> the subset of $\\Omega$ given by:</p>\n<p>$$<br>\\begin{equation*}{x\\in B}={\\omega \\in \\Omega ; X(\\omega )\\in B}\\end{equation*}<br>$$</p>\n<p>is in the $\\sigma$-algebra F</p>\n<p>本质是把事件映射为实数，同时为了保证可以拟映射，要求函数可测。</p>\n<p>构造R的Borel子集？从所有的闭区间出发，闭区间的交——特别地，开区间也包含进来，从而开集包含进来 ，因为每个开集可以写成开区间序列的并。闭集也是Borel集合，因为是开集的补集。</p>\n<p>关注X取值包含于某集合而不是具体的值</p>\n<p>Definition: let X be a random variable on a probability space, the distribution measure of X is the probability measure $\\mu _X$that assigns to each Borel subset B of R the mass $\\mu _X(B)=P{X\\in B}$ </p>\n<p>Random variable 有distribution 但两者不等同，两个不同的Random variable 可以有相同的distribution，一个单独的random variable 可以有两个不同的distribution</p>\n<p>cdf：$F(x)=P{X\\le x}$</p>\n<p>$\\mu_X(x,y]=F(y)-F(x)$</p>\n<h2 id=\"Expectations\"><a href=\"#Expectations\" class=\"headerlink\" title=\"Expectations\"></a>Expectations</h2><p>$E(X)=\\sum X(\\omega )P(\\omega )$</p>\n<p>${\\Omega , F, P}$ random variable $X(\\omega )$ P是概率空间中的测度</p>\n<p>$A_k={\\omega \\in \\Omega ; y_k\\le X(\\omega ) < y_{k+1}}$</p>\n<p>lower Lebesgue sum $LS^-_{\\Pi}=\\sum y_k P(A_k)$</p>\n<p>the maximal distance between the $y_k$ partition points approaches zero, we get Lebesgue integral$\\int_{\\Omega}X(\\omega )dP(\\omega)$</p>\n<p>Lebesgue integral相当于把积分概念拓展到可测空间，而不是单纯的更换了求和方式。横坐标实际上是$\\Omega$的测度。</p>\n<p>if the random variables X can take both positive and negative values, we can define the positive and negative parts of X:</p>\n<p>$X^+=max{X,0}$, $X^-=min{-X,0}$</p>\n<p>$\\int XdP=\\int X^+dP-\\int X^- dP$</p>\n<p><strong>Comparison</strong></p>\n<p>If $X\\le Y$ almost surely, and if the Lebesgue integral are defined, then</p>\n<p>$\\int _{\\Omega}X(\\omega)dP(\\omega)\\le \\int _{\\Omega}Y(\\omega)dP(\\omega)$</p>\n<p>If $X=Y$ almost surely, and if the Lebesgue integral are defined, then</p>\n<p>$\\int _{\\Omega}X(\\omega)dP(\\omega)=\\int _{\\Omega}Y(\\omega)dP(\\omega)$</p>\n<p><strong>Integrability, Linearity …</strong></p>\n<p><strong>Jensen’s inequality</strong></p>\n<p>if $\\phi$ is a convex, real-valued function defined on R and if $E|X|<\\infty$ ,then</p>\n<p>$\\phi(EX)\\le E\\phi(X)$</p>\n<p>$\\mathcal{B} (\\mathbb{R})$ be the sigma-algebra of Borel subsets of $\\mathbb{R}$, the Lebesgue measure $\\mathcal{L}$ on R assigns to each set $B\\in \\mathcal{B}(\\mathbb{R})$ a number in $[0,\\infty)$ or the value $\\infty$ so that:</p>\n<ol>\n<li>$\\mathcal{L}[a,b] =b-a$ </li>\n<li>if $B_1,B_2,B_3 \\dots$ is a sequence of disjoint sets in $\\mathcal{B}$ ,then we have the countable additivity property: $\\mathcal{L} (\\cup_{n=1}^\\infty B_n)=\\sum_{n=1}^\\infty \\mathcal{L} (B_n)$</li>\n</ol>\n<p>黎曼积分有且只有在区间中非连续点的集合的Lebesgue测度为零时有定义，即f在区间上几乎处处连续</p>\n<p>若f的Riemann积分在区间上存在，则f是Borel可测的，而且Riemann积分和Lebesgue积分一致。</p>\n<h3 id=\"convergence-of-integrals\"><a href=\"#convergence-of-integrals\" class=\"headerlink\" title=\"convergence of integrals\"></a>convergence of integrals</h3><p><strong>definition</strong></p>\n<p>$X_1,X_2,X_3\\dots$ be a sequence of random variables on the same probability space$(\\Omega, \\mathcal{F},\\mathbb{P})$, $X$ be another random variable. $X_1,X_2,X_3\\dots$ converges to $X$ almost surely $lim_{n\\to \\infty}X_n=X$ almost surely. if the set of $\\omega \\in \\Omega$ for the sequence has limit $X(\\omega)$ is a set with probability one. </p>\n<p>Strong law of Large Numbers: </p>\n<p>实的Borel可测的函数列$f_1,f_2,f_3\\dots$ defined on$\\mathbb{R}$, $f$ 也是实的Borel可测函数，the sequence converges to f almost every-where if 序列极限不为f的点的集合的 Lebesgue measure为零</p>\n<p>$lim_{n\\to \\infty}f_n=f \\textit{ almost everywhere}$</p>\n"}],"PostAsset":[],"PostCategory":[{"post_id":"ckypkquy000033opk9jpicgn7","category_id":"ckypkquy300043opkd6wnhh8i","_id":"ckypkquy700073opk24lhf6ey"}],"PostTag":[{"post_id":"ckypkquy000033opk9jpicgn7","tag_id":"ckypkquy500053opkaz7jbhv0","_id":"ckypkquy700083opkem7c8ght"},{"post_id":"ckypkquy000033opk9jpicgn7","tag_id":"ckypkquy600063opk91a4docu","_id":"ckypkquy700093opkel3efoqv"}],"Tag":[{"name":"金融","_id":"ckypkquy500053opkaz7jbhv0"},{"name":"数学","_id":"ckypkquy600063opk91a4docu"}]}}