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Motion in a random dynamical system can be informally thought of as a state evolving according to a succession of maps randomly chosen according to the distribution Q. [1] An example of a random dynamical system is a stochastic differential equation; in this case the distribution Q is typically determined by noise terms.
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 ...
Consider a probability space (Ω, Σ, P) and suppose that the (random) state Y t in n-dimensional Euclidean space R n of a system of interest at time t is a random variable Y t : Ω → R n given by the solution to an Itō stochastic differential equation of the form
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison ...
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have relevance to quantum field theory, statistical mechanics, and spatial modeling. [1] [2]
In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations to stochastic differential equations named after Leonhard Euler and Gisiro Maruyama. The ...
The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.
The LL method has been developed for a variety of equations such as the ordinary, delayed, random and stochastic differential equations. The LL integrators are key component in the implementation of inference methods for the estimation of unknown parameters and unobserved variables of differential equations given time series of (potentially ...