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Filtration (probability Theory)
In the theory of stochastic processes, a subdiscipline of probability theory, filtrations are totally ordered collections of subsets that are used to model the information that is available at a given point and therefore play an important role in the formalization of random (stochastic) processes. Definition Let (\Omega, \mathcal A, P) be a probability space and let I be an index set with a total order \leq (often \N , \R^+ , or a subset of \mathbb R^+ ). For every i \in I let \mathcal F_i be a sub-''σ''-algebra of \mathcal A . Then : \mathbb F:= (\mathcal F_i)_ is called a filtration, if \mathcal F_k \subseteq \mathcal F_\ell for all k \leq \ell . So filtrations are families of ''σ''-algebras that are ordered non-decreasingly. If \mathbb F is a filtration, then (\Omega, \mathcal A, \mathbb F, P) is called a filtered probability space. Example Let (X_n)_ be a stochastic process In probability theory and related fields, a stochastic () or rand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a . A probability space consists of three elements:Stroock, D. W. (1999). Probability theory: an analytic view. Cambridge University Press. # A '' sample space'', \Omega, which is the set of all possible outcomes of a random process under consideration. # An event space, \mathcal, which is a set of events, where an event is a subset of outcomes in the sample space. # A '' probability function'', P, which assigns, to each event in the event space, a probability, which is a number between 0 and 1 (inclusive). In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, # The sample space \Omega is typically the set \ where each element in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an ''indexed family'', often written as . Examples *An enumeration of a set gives an index set J \sub \N, where is the particular enumeration of . *Any countably infinite set can be (injectively) indexed by the set of natural numbers \N. *For r \in \R, the indicator function on is the function \mathbf_r\colon \R \to \ given by \mathbf_r (x) := \begin 0, & \mbox x \ne r \\ 1, & \mbox x = r. \end The set of all such indicator functions, \_, is an uncountable set indexed by \mathbb. Other uses In computational complexity theory and cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events with a well-defined probability. In this way, σ-algebras help to formalize the notion of ''size''. In formal terms, a σ-algebra (also σ-field, where the σ comes from the German "Summe", meaning "sum") on a set ''X'' is a nonempty collection Σ of subsets of ''X'' closed under complement, countable unions, and countable intersections. The ordered pair (X, \Sigma) is called a measurable space. The set ''X'' is understood to be an ambient space (such as the 2D plane or the set of outcomes when rolling a six-sided die ), and the collection Σ is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to cap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Filtration
In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration. Formal definition Let * (\Omega, \mathcal A, P) be a probability space. *(I, \le) be a totally ordered index set. In many examples, the index set I is the natural numbers \N (possibly including 0) or an interval , T/math> or , +\infty). *(S, \Sigma) be a measurable space. Often, the state space S is often the real line \R or Euclidean space \R^n. *X:I\times\Omega\to S be a stochastic process. Then the natural filtration of \mathcal with respect to X is defined to be the filtration \mathbb^X=\mathcal^X_\bullet=(\mathcal^X_i)_ given by :\mathcal^X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Set
In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S \in \Sigma such that \mu(S) = 0. Examples Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers , the set of rational numbers and the set of algebraic numbers are all countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Measure Space
In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is complete if and only if :S \subseteq N \in \Sigma \mbox \mu(N) = 0\ \Rightarrow\ S \in \Sigma. Motivation The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by (\R, B, \lambda). We now wish to construct some two-dimensional Lebesgue measure \lambda^2 on the plane \R^2 as a product measure. Naively, we would take the -algebra on \R^2 to be B \otimes B, the smallest -algebra containing all measurable "rectangles" A_1 \times A_2 for A_1, A_2 \in B. While this approach does define a measure space, it has a flaw. Since every singleton set has one-dimensional Lebesgue measure zero, \la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Filtration
In the theory of stochastic processes in mathematics and statistics, the generated filtration or natural filtration associated to a stochastic process is a filtration associated to the process which records its "past behaviour" at each time. It is in a sense the simplest filtration available for studying the given process: all information concerning the process, and only that information, is available in the natural filtration. Formal definition Let * (\Omega, \mathcal A, P) be a probability space. *(I, \le) be a totally ordered index set. In many examples, the index set I is the natural numbers \N (possibly including 0) or an interval , T/math> or , +\infty). *(S, \Sigma) be a measurable space. Often, the state space S is often the real line \R or Euclidean space \R^n. *X:I\times\Omega\to S be a stochastic process. Then the natural filtration of \mathcal with respect to X is defined to be the filtration \mathbb^X=\mathcal^X_\bullet=(\mathcal^X_i)_ given by :\mathcal^X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
