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Increasing Process
An increasing process is a stochastic process :(X_t)_ where the random variables X_t which make up the process are increasing almost surely and adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...: :0=X_0 \leq X_ \leq \cdots . A continuous increasing process is such a process where the set M is continuous. Stochastic processes {{probability-stub ... [...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] |
Random Variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost surely'': an event that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adapted Process
In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every realisation and every ''n'', ''Xn'' is known at time ''n''. The concept of an adapted process is essential, for instance, in the definition of the Itō integral, which only makes sense if the integrand is an adapted process. Definition Let * (\Omega, \mathcal, \mathbb) be a probability space; * I be an index set with a total order \leq (often, I is \mathbb, \mathbb_0, , T/math> or filtration of the sigma algebra \mathcal; * (S,\Sigma) be a measurable space, the ''state space''; * X: I \times \Omega \to S be a stochastic process. The process X is said to be adapted to the filtration \left(\mathcal_i\right)_ if the random variable X_i: \Omega \to S is a (\mathcal_i, \Sigma)-measurable function for each i \in I. Examples Consider a stochastic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
