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Killed Process
In probability theory — specifically, in stochastic analysis — a killed process is a stochastic process that is forced to assume an undefined or "killed" state at some (possibly random) time. Definition Let ''X'' : ''T'' × Ω → ''S'' be a stochastic process defined for "times" ''t'' in some ordered index set ''T'', on a probability space (Ω, Σ, P), and taking values in a measurable space ''S''. Let ''ζ'' : Ω → ''T'' be a random time, referred to as the killing time. Then the killed process ''Y'' associated to ''X'' is defined by :Y_ = X_ \mbox t < \zeta, and ''Y''''t'' is left undefined for ''t'' ≥ ''ζ''. Alternatively, one may set ''Y''''t'' = ''c'' for ''t'' ≥ ''ζ'', where ''c'' is a "coffin state" not in ''S''. See also * |
Probability Theory
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Processes
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. 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, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...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, an index set is a set for which there exists an algorithm that can sample the set efficiently; e. ... [...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 die. 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. # An event space, which is a set of events \mathcal, an event being a set of outcomes in the sample space. # A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be \. For the event space, we could simply use the set of all subsets of the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the tuple (X, \mathcal A) is called a measurable space. Note that in contrast to a measure space, no measure is needed for a measurable space. Example Look at the set: X = \. One possible \sigma-algebra would be: \mathcal A_1 = \. Then \left(X, \mathcal A_1\right) is a measurable space. Another possible \sigma-algebra would be the power set on X: \mathcal A_2 = \mathcal P(X). With this, a second measurable space on the set X is given by \left(X, \mathcal A_2\right). Common measurable spaces If X is finite or countably infinite, the \sigma-algebra is most often the power set on X, so \mathcal A = \mathcal P(X). This leads to the measurable space (X, \mathcal P(X)). If X is a topological space In mathematics, a topological space is, rou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stopped Process
In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * (\mathbb, \mathcal) be a measurable space; * X : ,_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty/math>_be_a_stopping_rule">stopping_time_with_respect_to_some__be_a_stopping_rule.html"__"title=",_+_\infty.html"_;"title=",_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty">,_+_\infty)_\times_\Omega_\to_\mathbb_be_a_stochastic_process; *_\tau_:_\Omega_\to_[0,_+_\infty/math>_be_a_stopping_rule">stopping_time_with_respect_to_some_filtration_(abstract_algebra)">filtration_\_of_\mathcal. Then_the_stopped_process_X^_is_defined_for_t_\geq_0_and_\omega_\in_\Omega_by :X_^_(\omega)_:=_X__(\omega). _Examples _Gambling Consider_a_Gambling.html" "title="filtration_(abstract_algebra).html" "tit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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