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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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. 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 process, ... [...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] |
Stopping Rule
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time. Stopping times occur in decision theory, and the optional stopping theorem is an important result in this context. Stopping times are also frequently applied in mathematical proofs to “tame the continuum of time”, as Chung put it in his book (1982). Definition Discrete time Let \tau be a random variable, which is defined on the filtered probability space (\Omega, \mathcal F, (\mathcal F_n)_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filtration (abstract Algebra)
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k is an algebra (A,\cdot) over k that has an increasing sequence \ \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A of subspaces of A such that :A=\bigcup_ F_ and that is compatible with the multiplication in the following sense: : \forall m,n \in \mathbb,\quad F_m\cdot F_n\subseteq F_. Associated graded algebra In general there is the following construction that produces a graded algebra out of a filtered algebra. If A is a filtered algebra then the ''associated graded algebra'' \mathcal(A) is defined as follows: The multiplication is well-defined and endows \mathcal(A) with the structure of a graded algebra, with gradation \_. Furthermore if A is associative then so is \mathcal(A). Also if A is uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three elements to be present: consideration (an amount wagered), risk (chance), and a prize. The outcome of the wager is often immediate, such as a single roll of dice, a spin of a roulette wheel, or a horse crossing the finish line, but longer time frames are also common, allowing wagers on the outcome of a future sports contest or even an entire sports season. The term "gaming" in this context typically refers to instances in which the activity has been specifically permitted by law. The two words are not mutually exclusive; ''i.e.'', a "gaming" company offers (legal) "gambling" activities to the public and may be regulated by one of many gaming control boards, for example, the Nevada Gaming Control Board. However, this distinction is not u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roulette
Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the color red or black, whether the number is odd or even, or if the numbers are high (19–36) or low (1–18). To determine the winning number, a croupier spins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular track running around the outer edge of the wheel. The ball eventually loses momentum, passes through an area of deflectors, and falls onto the wheel and into one of thirty-seven (single-zero, French or European style roulette) or thirty-eight (double-zero, American style roulette) or thirty-nine (triple-zero, "Sands Roulette") colored and numbered pockets on the wheel. The winnings are then paid to anyone who has placed a successful bet. History The first form of roulette was devised in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casino
A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known for hosting live entertainment, such as stand-up comedy, concerts, and sports. and usage ''Casino'' is of Italian origin; the root means a house. The term ''casino'' may mean a small country villa, summerhouse, or social club. During the 19th century, ''casino'' came to include other public buildings where pleasurable activities took place; such edifices were usually built on the grounds of a larger Italian villa or palazzo, and were used to host civic town functions, including dancing, gambling, music listening, and sports. Examples in Italy include Villa Farnese and Villa Giulia, and in the US the Newport Casino in Newport, Rhode Island. In modern-day Italian, a is a brothel (also called , literally "closed house"), a mess (confusing situation), or a noisy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brownian Motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem). This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking throu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hitting Time
In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times. Definitions Let ''T'' be an ordered index set such as the natural numbers, N, the non-negative real numbers, , +∞), or a subset of these; elements ''t'' ∈ ''T'' can be thought of as "times". Given a probability space (Ω, Σ, Pr) and a measurable space">measurable state space ''S'', let ''X'' : Ω × ''T'' → ''S'' be a stochastic process, and let ''A'' be a measurable set, measurable subset of the state space ''S''. Then the first hit time ''τ''''A'' : Ω → [0, +∞] is the random variable defined by :\tau_A (\omega) := \inf \. The first exit time (from ''A'') is defined to be the first hit time for ''S'' \ ''A'', the complement (set theory), c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 * |
