Editing Stochastic differential equation (section)

==Background==
Stochastic differential equations originated in the theory of [[Brownian motion]], in the work of  [[Annus Mirabilis Papers#Brownian motion|Albert Einstein]] and  [[Marian Smoluchowski#Work|Marian Smoluchowski]] in 1905, although  [[Louis Bachelier]] was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known as [[Bachelier model]]. Some of these early examples were linear stochastic differential equations, also called [[Langevin equation]]s after French physicist [[Paul Langevin|Langevin]], describing the motion of a harmonic oscillator subject to a random force. 
The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician [[Kiyosi Itô]], who introduced the concept of [[stochastic integral]] and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist [[Ruslan L. Stratonovich|Stratonovich]], leading to a calculus similar to ordinary calculus. 

===Terminology===
The most common form of SDEs in the literature is an [[ordinary differential equation]] with the right hand side perturbed by a term dependent on a [[white noise]] variable. In most cases, SDEs are understood as continuous time limit of the corresponding [[stochastic difference equation]]s. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral.<ref name="rogerswilliams"/><ref name="oksendal"/> Such a mathematical definition was first proposed by [[Kiyosi Itô]] in the 1940s, leading to what is known today as the [[Itô calculus]].
Another construction was later proposed by Russian physicist [[Ruslan L. Stratonovich|Stratonovich]], 
leading to what is known as the [[Stratonovich integral]].
The [[Itô integral]] and [[Stratonovich integral]] are related, but different, objects and the choice between them depends on the application considered. The [[Itô calculus]] is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time.
The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion on [[manifolds]], although it is possible and in some cases preferable to model random motion on manifolds through Itô SDEs,<ref name="Emery"/> for example when trying to optimally approximate SDEs on submanifolds.<ref name="armstrongprojection">Armstrong, J., Brigo, D. and Rossi Ferrucci, E. (2019), Optimal approximation of SDEs on submanifolds: the Itô-vector and Itô-jet projections. Proc. London Math. Soc., 119: 176-213. https://doi.org/10.1112/plms.12226.</ref> 

An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the [[Smoluchowski equation]] or the [[Fokker–Planck equation]], an equation describing the time evolution of [[Probability density function|probability distribution function]]s. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of [[Supersymmetric theory of stochastic dynamics#Stochastic evolution operator|stochastic evolution operator]].

In physical science, there is an ambiguity in the usage of the term [[Langevin equation|"Langevin SDEs"]]. While Langevin SDEs can be of a [[Langevin equation#Generic Langevin equation|more general form]], this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure,<ref>{{Cite journal|last1=Parisi|first1=G.|last2=Sourlas|first2=N.|date=1979|title=Random Magnetic Fields, Supersymmetry, and Negative Dimensions |journal=Physical Review Letters|volume=43|issue=11|pages=744–745|doi=10.1103/PhysRevLett.43.744|bibcode=1979PhRvL..43..744P}}</ref> leading to a N=2 supersymmetric model closely related to [[supersymmetric quantum mechanics]]. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e., [[Supersymmetric theory of stochastic dynamics#Spontaneous supersymmetry breaking and chaos|(overdamped) Langevin SDEs are never chaotic]].

===Stochastic calculus===
[[Brownian motion]] or the [[Wiener process]] was discovered to be exceptionally complex mathematically. The [[Wiener process]] is almost surely nowhere differentiable;<ref name="rogerswilliams"/><ref name="oksendal"/> thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the [[Itô calculus|Itô stochastic calculus]] and the [[Stratonovich stochastic calculus]]. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003)<ref name="oksendal">{{cite book
| last = Øksendal
| first = Bernt K.
| author-link = Bernt Øksendal
| title=Stochastic Differential Equations: An Introduction with Applications
| publisher=Springer
| location = Berlin
| year=2003
| isbn=3-540-04758-1
}}</ref> and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again.<ref name="rogerswilliams"/><ref name="oksendal"/> Still, one must be careful which calculus to use when the SDE is initially written down.

===Numerical solutions===
Numerical methods for solving stochastic differential equations<ref name="kloeden">Kloeden, P.E., Platen E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin, Heidelberg. DOI: https://doi.org/10.1007/978-3-662-12616-5</ref> include the [[Euler–Maruyama method]], [[Milstein method]], [[Runge–Kutta method (SDE)]], Rosenbrock method,<ref name="Averina">Artemiev, S.S., Averina, T.A. (1997). Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, Utrecht, The Netherlands. DOI: https://doi.org/10.1515/9783110944662</ref> and methods based on different representations of iterated stochastic integrals.<ref name="Kuznetsov">Kuznetsov, D.F. (2023). Strong approximation of iterated Itô and Stratonovich stochastic integrals: Method of generalized multiple Fourier series. Application to numerical integration of Itô SDEs and semilinear SPDEs. Differ. Uravn. Protsesy Upr., no. 1. DOI: https://doi.org/10.21638/11701/spbu35.2023.110</ref><ref name="Rybakov">Rybakov, K.A. (2023). Spectral representations of iterated stochastic integrals and their application for modeling nonlinear stochastic dynamics. Mathematics, vol. 11, 4047. DOI: https://doi.org/10.3390/math11194047</ref>