Editing Stochastic differential equation (section)

{{Short description|Differential equations involving stochastic processes}}
{{Differential equations}}

A '''stochastic differential equation''' ('''SDE''') is a [[differential equation]] in which one or more of the terms is a [[stochastic process]],<ref name="rogerswilliams">{{Cite book|last1=Rogers |first1=L.C.G. |author-link1=Chris Rogers (mathematician)|last2=Williams | first2 = David| author-link2=David Williams (mathematician)| edition= 2nd ed., Cambridge Mathematical Library|title=Diffusions, Markov Processes and Martingales, Vol 2: Ito Calculus |publisher=[[Cambridge University Press]] | year=2000 | isbn=0-521-77594-9 | oclc=42874839 | doi= 10.1017/CBO9780511805141 }}</ref> resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to [[mathematical model|model]] various behaviours of stochastic models such as [[stock price]]s,<ref name="musielarutkowski">Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.</ref> [[random growth model]]s<ref name="oksendal"/> or physical systems that are subjected to [[thermal fluctuations]].

SDEs have a random differential that is in the most basic case random [[white noise]] calculated as the distributional derivative of a [[Brownian motion]] or more generally a [[semimartingale]]. However, other types of random behaviour are possible, such as [[jump process]]es like [[Lévy process]]es<ref>Kunita, H. (2004). Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2054-1_6</ref> or semimartingales with jumps. 

Stochastic differential equations are in general neither differential equations nor [[random differential equations]]. Random differential equations are conjugate to stochastic differential equations. Stochastic differential equations can also be extended to [[differential manifold]]s.<ref>{{Cite journal|last1=Imkeller|first1=Peter|last2=Schmalfuss|first2=Björn|date=2001|title=The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors|url=http://dx.doi.org/10.1023/a:1016673307045|journal=Journal of Dynamics and Differential Equations|volume=13|issue=2|pages=215–249|doi=10.1023/a:1016673307045|s2cid=3120200|issn=1040-7294}}</ref><ref name="Emery">Michel Emery (1989). Stochastic calculus in manifolds. Springer Berlin, Heidelberg. Doi  https://doi.org/10.1007/978-3-642-75051-9</ref><ref>Zdzisław Brzeźniak and [[K._David_Elworthy|K. D. Elworthy]], Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology 6 (2000), no. 1, 43-84.</ref><ref name="sdesjets">Armstrong J. and [[Damiano_Brigo|Brigo D.]] (2018). Intrinsic stochastic differential equations as jets. Proc. R. Soc. A., 474: 20170559, 
http://doi.org/10.1098/rspa.2017.0559</ref>