In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes.[ clarification needed ]
Given a filtered probability space , then a stochastic process is predictable if is measurable with respect to the σ-algebra for each n. [1]
Given a filtered probability space , then a continuous-time stochastic process is predictable if , considered as a mapping from , is measurable with respect to the σ-algebra generated by all left-continuous adapted processes. [2] This σ-algebra is also called the predictable σ-algebra.
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.
In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov.
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.
In mathematics, a filtration is an indexed family of subobjects of a given algebraic structure , with the index running over some totally ordered index set , subject to the condition that
In probability theory, in particular in the study of stochastic processes, a stopping 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.
Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion. It has important applications in mathematical finance and stochastic differential equations.
In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”
In the study of stochastic processes, an adapted 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.
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. Progressively measurable processes are important in the theory of Itô integrals.
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure onto a finite-dimensional vector space.
In mathematics, the Kolmogorov extension theorem is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov..
In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed time.
In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.
In probability theory, a Markov kernel is a map that in the general theory of Markov processes, plays the role that the transition matrix does in the theory of Markov processes with a finite state space.
In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.
The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.
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 processes.
