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In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition. [1] The problem is named after Anatoliy Skorokhod who first published the solution to a stochastic differential equation for a reflecting Brownian motion. [2] [3] [4]
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
However, when applied to a discrete and finite process () {, …,}, the problem takes a much simpler form and standard algebra can be used to carry out the calculations. Note that a continuous process can also be sampled at N points in time in order to reduce the problem to a finite version.
In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter.. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to anal
In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds.It is therefore a synthesis of stochastic analysis (the extension of calculus to stochastic processes) and of differential geometry.
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but ...
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, [1] resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices , [ 2 ] random ...