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→‎斯科罗霍德空间 Skorokhod space

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====斯科罗霍德空间 Skorokhod space====

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A '''Skorokhod space''', also written as '''Skorohod space''', is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as <math>[0,1]</math> or <math>[0,\infty)</math>, and take values on the real line or on some metric space. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression ''continue à droite, limite à gauche'', due to the functions being right-continuous with left limits. A Skorokhod function space, introduced by [[Anatoliy Skorokhod]], is often denoted with the letter <math>D</math>,<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/> so the function space is also referred to as space <math>D</math>. The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the [[unit interval]] <math>[0,1]</math>.<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>

''skorokod space''也写为''Skorohod space''，是所有右连续左极限的函数的数学空间，定义在实线的某个区间上，例如<math>[0,1]</math>或<math>[0，\infty）</math>，取实线或度量空间上的值。<ref name="Whitt2006page78">{{cite book|author=Ward Whitt|title=Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues|url=https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21748-2|pages=78–79}}</ref><ref name="GusakKukush2010page24">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 24</ref><ref name="Bogachev2007Vol2page53">{{cite book|author=Vladimir I. Bogachev|title=Measure Theory (Volume 2)|url=https://books.google.com/books?id=CoSIe7h5mTsC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-34514-5|page=53}}</ref>这些函数被称为cádLag或cadlag函数，这是基于法语表达式“continue a droite，limiteégauche”的首字母缩略词，因为这些函数是右连续的，具有左极限。<ref name="Whitt2006page78"/><ref name="Klebaner2005page4">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=4}}</ref>由[[Anatoliy Skorokod]]引入的Skorokod函数空间，<ref name="Bogachev2007Vol2page53"/>通常用字母<math>D</math>表示，<ref name="Whitt2006page78"/><ref name="GusakKukush2010page24"/><ref name="Bogachev2007Vol2page53"/><ref name="Klebaner2005page4"/>因此函数空间也被称为空间<math>D</math><ref name="Whitt2006page78"/><ref name="Asmussen2003page420">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=420}}</ref><ref name="Billingsley2013page121">{{cite book|author=Patrick Billingsley|title=Convergence of Probability Measures|url=https://books.google.com/books?id=6ItqtwaWZZQC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-62596-5|page=121}}</ref>此函数空间的表示法还可以包括定义所有cádlág函数的间隔，因此，例如，<math>D[0,1]</math>表示在单位间隔<math>[0，1] </math>。<ref name="Klebaner2005page4"/><ref name="Billingsley2013page121"/><ref name="Bass2011page34">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=34}}</ref>

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在随机过程理论中，由于通常假定连续时间随机过程的样本函数属于一个Skorokod空间，<ref name="Bogachev2007Vol2page53"/><ref name="Asmussen2003page420"/>因此经常使用Skorokod函数空间，对应于Wiener过程的样本函数。但是空间也有间断函数，这意味着随机过程的样本函数具有跳跃性，例如泊松过程（在实线上），同时也是这一领域的成员。<ref name="Billingsley2013page121"/><ref name="BinghamKiesel2013page154">{{cite book|author1=Nicholas H. Bingham|author2=Rüdiger Kiesel|title=Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives|url=https://books.google.com/books?id=AOIlBQAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-3856-3|page=154}}</ref>

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