Editing Stochastic differential equation

{{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>

==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>

==Use in physics==
{{See also|Langevin equation}}
In physics, SDEs have wide applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the [[dynamical systems theory]] to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.

There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:

:<math id="genSDE">\frac{\mathrm{d}x(t)}{\mathrm{d}t} = F(x(t)) + \sum_{\alpha=1}^ng_\alpha(x(t))\xi^\alpha(t),\,</math>

where <math>x\in X </math> is the position in the system in its [[phase space|phase (or state) space]], <math>X</math>, assumed to be a differentiable manifold, the <math>F\in TX</math> is a flow vector field representing deterministic law of evolution, and <math>g_\alpha\in TX </math> is a set of vector fields that define the coupling of the system to Gaussian white noise, <math>\xi^\alpha</math>. If <math> X </math> is a linear space and <math>g</math> are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. For additive noise, the Itô and Stratonovich forms of the SDE generate the same solution, and it is not important which definition is used to solve the SDE. For multiplicative noise SDEs the Itô and Stratonovich forms of the SDE are different, and care should be used in mapping between them.<ref>{{Cite book| edition = 4th| publisher = Springer-Verlag| isbn = 978-3-540-70712-7| last = Gardiner| first = C W| title = Handbook of Stochastic Methods| location = Berlin| year = 2009}}</ref>

For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.<ref>{{Cite journal|last=Slavík|first=A.|title=Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters|journal=Journal of Mathematical Analysis and Applications|language=en|volume=402|issue=1|pages=261–274|doi=10.1016/j.jmaa.2013.01.027|year=2013|doi-access=free}}</ref> Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a [[stochastic difference equation]]. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a [[Supersymmetric theory of stochastic dynamics#Resolution of Ito–Stratonovich dilemma and operator ordering conventions|uniquely defined mathematical object]] that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.

In physics, the main method of solution is to find the [[probability distribution]] function as a function of time using the equivalent [[Fokker–Planck equation]] (FPE). The Fokker–Planck equation is a deterministic [[partial differential equation]]. It tells how the probability distribution function evolves in time similarly to how the  [[Schrödinger equation]] gives the time evolution of the quantum wave function or the [[diffusion equation]] gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by [[Monte Carlo Method|Monte Carlo]] simulation. Other techniques include the [[path integral formulation|path integration]] that draws on the analogy between statistical physics and [[quantum mechanics]] (for example, the Fokker-Planck equation can be transformed into the [[Schrödinger equation]] by rescaling a few variables) or by writing down [[ordinary differential equations]] for the statistical [[moment (mathematics)|moments]] of the probability distribution function. {{Citation needed|date=August 2011}}

==Use in probability and mathematical finance==
The notation used in [[probability theory]] (and in many applications of probability theory, for instance in signal processing with the [[Filtering problem (stochastic processes)|filtering problem]] and in [[mathematical finance]]) is slightly different. It is also the notation used in publications on [[numerical methods]] for solving stochastic differential equations. This notation makes the exotic nature of the random function of time <math>\xi^\alpha</math> in the physics formulation more explicit. In strict mathematical terms,  <math>\xi^\alpha</math> cannot be chosen as an ordinary function, but only as a [[generalized function]]. The mathematical formulation treats this complication with less ambiguity than the physics formulation.

A typical equation is of the form

:<math> \mathrm{d} X_t = \mu(X_t,t)\, \mathrm{d} t +  \sigma(X_t,t)\, \mathrm{d} B_t , </math>

where <math>B</math> denotes a [[Wiener process]] (standard Brownian motion).
This equation should be interpreted as an informal way of expressing the corresponding [[integral equation]]

:<math> X_{t+s} - X_{t} = \int_t^{t+s} \mu(X_u,u) \mathrm{d} u + \int_t^{t+s} \sigma(X_u,u)\, \mathrm{d} B_u . </math>

The equation above characterizes the behavior of the [[continuous time]] [[stochastic process]] ''X''<sub>''t''</sub> as the sum of an ordinary [[Lebesgue integral]] and an [[Itô calculus|Itô integral]]. A [[heuristic]] (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length ''δ'' the stochastic process ''X''<sub>''t''</sub> changes its value by an amount that is [[normal distribution|normally distributed]] with [[expected value|expectation]] ''μ''(''X''<sub>''t''</sub>,&nbsp;''t'')&nbsp;''δ'' and [[variance]] ''σ''(''X''<sub>''t''</sub>,&nbsp;''t'')<sup>2</sup>&nbsp;''δ'' and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function ''μ'' is referred to as the drift coefficient, while ''σ'' is called the diffusion coefficient. The stochastic process ''X''<sub>''t''</sub> is called a [[diffusion process]], and satisfies the [[Markov property]].<ref name="rogerswilliams" />

The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution<ref name="rogerswilliams" /> Both require the existence of a process ''X''<sub>''t''</sub> that solves the integral equation version of the SDE. The difference between the two lies in the underlying [[probability space]] (<math>\Omega,\, \mathcal{F},\, P</math>). A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space. The [[Yamada–Watanabe theorem]] makes a connection between the two.

An important example is the equation for [[geometric Brownian motion]]

:<math>\mathrm{d} X_t = \mu X_t \, \mathrm{d} t +  \sigma X_t \, \mathrm{d} B_t.</math>

which is the equation for the dynamics of the price of a [[stock]] in the [[Black–Scholes model|Black–Scholes]] options pricing model<ref name="musielarutkowski" /> of financial mathematics.

Generalizing the geometric Brownian motion, it is also possible to define SDEs admitting strong solutions and whose distribution is a convex combination of densities coming from different geometric Brownian motions or Black Scholes models, obtaining  a single SDE whose solutions is distributed as a mixture dynamics of lognormal  distributions of different Black Scholes models.<ref name="musielarutkowski"/><ref>Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI https://doi.org/10.1007/3-540-30591-2</ref><ref>{{cite journal | title = Lognormal-mixture dynamics and calibration to market volatility smiles | first1 = Damiano | last1 = Brigo | author-link1 = Damiano Brigo | first2 = Fabio | last2 = Mercurio | author-link2 = Fabio Mercurio | pages = 427–446 | journal = International Journal of Theoretical and Applied Finance | volume = 5 | year = 2002 | issue = 4 | doi = 10.1142/S0219024902001511 }}</ref><ref>Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, {{ISSN|1469-7688}}</ref> This leads to models that can deal with the [[volatility smile]] in financial mathematics.

The simpler SDE called [[Geometric Brownian motion#Arithmetic Brownian Motion|arithmetic Brownian motion]]<ref name="oksendal"/>  

:<math>\mathrm{d} X_t = \mu \, \mathrm{d} t +  \sigma  \, \mathrm{d} B_t</math>

was used by Louis Bachelier as the first model for stock prices in 1900, known today as [[Bachelier model]].
 
There are also more general stochastic differential equations where the coefficients ''μ'' and ''σ'' depend not only on the present value of the process ''X''<sub>''t''</sub>, but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process, ''X'', is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values of ''X'', the defining equation is called a stochastic delay differential equation.

A generalization of stochastic differential equations with the Fisk-Stratonovich integral to semimartingales with jumps are the SDEs of ''Marcus type''. The Marcus integral is an extension of McShane's stochastic calculus.<ref>{{citation|author=Steven Marcus|date=1981|pages=223–245|periodical=Stochastics|title=Modeling and approximation of stochastic differential equation driven by semimartigales|volume=4}}<!-- auto-translated by Module:CS1 translator --></ref>

An innovative application in stochastic finance derives from the usage of the equation for [[Ornstein–Uhlenbeck process]]
 
:<math>\mathrm{d} R_t = \mu R_t \, \mathrm{d} t +  \sigma_t \, \mathrm{d} B_t.</math>
 
which is the equation for the dynamics of the return of the price of a [[stock]] under the hypothesis that returns display a [[Log-normal distribution]].
Under this hypothesis, the methodologies developed by Marcello Minenna determines prediction interval able to identify abnormal return that could hide [[market abuse]] phenomena. <ref>{{cite web |url=https://www.risk.net/regulation/1528679/detecting-market-abuse |title=Detecting Market Abuse |date=2 November 2004 | publisher=Risk Magazine}}</ref><ref>{{cite web |url=https://www.consob.it/documents/1912911/2006254/qdf54en.pdf/d31b160c-9ba5-e08d-d39d-a78bddfc698e |title=The detection of Market Abuse on financial markets: a quantitative approach | publisher=Consob – The Italian Securities and Exchange Commission}}</ref>

=== SDEs on manifolds ===
More generally one can extend the theory of stochastic calculus onto [[differential manifold]]s and for this purpose one uses the Fisk-Stratonovich integral. Consider a manifold <math>M</math>, some finite-dimensional vector space <math>E</math>, a filtered probability space <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in \R_{+}},P)</math> with <math>(\mathcal{F}_t)_{t\in \R_{+}}</math> satisfying the [[usual conditions]] and let <math>\widehat{M}=M\cup \{\infty\}</math> be the [[one-point compactification]] and <math>x_0</math> be <math>\mathcal{F}_0</math>-measurable. A ''stochastic differential equation on <math>M</math>'' written
:<math>\mathrm{d}X=A(X)\circ dZ</math>
is a pair <math>(A,Z)</math>, such that
*<math>Z</math> is a continuous <math>E</math>-valued semimartingale,
*<math>A:M\times E\to TM, (x,e)\mapsto A(x)e</math> is a homomorphism of [[vector bundle]]s over <math>M</math>.
For each <math>x\in M</math> the map <math>A(x):E\to T_{x}M</math> is linear and <math>A(\cdot)e\in \Gamma(TM)</math> for each <math>e\in E</math>. 

A solution to the SDE on <math>M</math> with initial condition <math>X_0=x_0</math> is a continuous <math>\{\mathcal{F}_t\}</math>-adapted <math>M</math>-valued process <math>(X_t)_{t<\zeta}</math> up to life time <math>\zeta</math>, s.t. for each test function <math>f\in C_c^{\infty}(M)</math> the process <math>f(X)</math> is a real-valued semimartingale and for each stopping time <math>\tau</math> with <math>0\leq \tau < \zeta</math> the equation
:<math>f(X_{\tau})=f(x_0)+\int_0^\tau (\mathrm{d}f)_X A(X)\circ \mathrm{d}Z</math>
holds <math>P</math>-almost surely, where <math>(df)_X:T_xM\to T_{f(x)}M</math> is the [[Differential form|differential]] at <math>X</math>. It is a ''maximal solution'' if the life time is maximal, i.e.,
:<math>\{\zeta <\infty\}\subset\left\{\lim\limits_{t\nearrow \zeta}X_t=\infty \text{ in }\widehat{M}\right\}</math>
<math>P</math>-almost surely. It follows from the fact that <math>f(X)</math> for each test function <math>f\in C_c^{\infty}(M)</math> is a semimartingale, that <math>X</math> is a ''semimartingale on <math>M</math>''. Given a maximal solution we can extend the time of <math>X</math> onto full <math>\R_+</math> and after a continuation of <math>f</math> on <math>\widehat{M}</math> we get
:<math>f(X_{t})=f(X_0)+\int_0^t (\mathrm{d}f)_X A(X)\circ \mathrm{d}Z, \quad t\geq 0</math>
up to indistinguishable processes.<ref>{{cite book|first1=Wolfgang|last1=Hackenbroch|first2=Anton|last2=Thalmaier|publisher=Vieweg+Teubner Verlag Wiesbaden|title=Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale|date=1994 |isbn=978-3-519-02229-9|page=364-365|lang=de}}</ref>
Although Stratonovich SDEs are the natural choice for SDEs on manifolds, given that they satisfy the chain rule and that their drift and diffusion coefficients behave as vector fields under changes of coordinates, there are cases where Ito calculus on manifolds is preferable. A theory of Ito calculus on manifolds was first developed by [[Laurent Schwartz]] through the concept of Schwartz morphism,<ref name="Emery"/> see also the related 2-jet interpretation of Ito SDEs on manifolds based on the jet-bundle.<ref name="sdesjets"/> This interpretation is helpful when trying to optimally approximate the solution of an SDE given on a large space with the solutions of an SDE given on a submanifold of that space,<ref name="armstrongprojection"/> in that a Stratonovich based projection does not  result to be optimal. This has been applied to the [[filtering problem]], leading to optimal projection filters.<ref name="armstrongprojection"/>

== As rough paths ==
Usually the solution of an SDE requires a probabilistic setting, as the integral implicit in the solution is a stochastic integral. If it were possible to deal with the differential equation path by path, one would not need to define a stochastic integral and one could develop a theory  independently of probability theory. 
This points to considering the SDE

:<math> \mathrm{d} X_t(\omega) = \mu(X_t(\omega),t)\, \mathrm{d} t +  \sigma(X_t(\omega),t)\, \mathrm{d} B_t(\omega) </math>

as a single deterministic differential equation for every <math>\omega \in \Omega</math>, where <math>\Omega</math> is the sample space in the given probability space (<math>\Omega,\, \mathcal{F},\, P</math>). However, a direct path-wise interpretation of the SDE is not possible, as the Brownian motion paths have unbounded variation and are nowhere differentiable with probability one, so that there is no naive way to give meaning to terms like <math>\mathrm{d} B_t(\omega)</math>, precluding also a naive path-wise definition of the stochastic integral as an integral against every single <math>\mathrm{d} B_t(\omega)</math>.  However, motivated by the Wong-Zakai result<ref name="frizhairer"/> for limits of solutions of SDEs with regular noise and using [[rough paths]] theory, while adding a chosen definition of iterated integrals of Brownian motion, it is possible to define a deterministic rough integral for every single <math>\omega \in \Omega</math> that coincides for example with the Ito integral with probability one for a particular choice of the iterated Brownian integral.<ref name="frizhairer">Friz, P. and Hairer, M. (2020). A Course on Rough Paths with an Introduction to Regularity Structures, 2nd ed., Springer-Verlag, Heidelberg, DOI 
 https://doi.org/10.1007/978-3-030-41556-3</ref> Other definitions of the iterated integral lead to deterministic pathwise equivalents of different stochastic integrals, like the Stratonovich integral. This has been used for example in financial mathematics to price options without probability.<ref name="optionswithoutprobability"> Armstrong, J., Bellani, C., Brigo, D. and Cass, T. (2021). Option pricing models without probability: a rough paths approach. Mathematical Finance, vol. 31, pages 1494–1521.</ref>

==Existence and uniqueness of solutions==

As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> and driven by an ''m''-dimensional Brownian motion ''B''; the proof may be found in Øksendal (2003, §5.2).<ref name="oksendal"/>

Let ''T''&nbsp;&gt;&nbsp;0, and let

:<math>\mu : \mathbb{R}^{n} \times [0, T] \to \mathbb{R}^{n};</math>
:<math>\sigma : \mathbb{R}^{n} \times [0, T] \to \mathbb{R}^{n \times m};</math>

be [[measurable function]]s for which there exist constants ''C'' and ''D'' such that

:<math>\big| \mu (x, t) \big| + \big| \sigma (x, t) \big| \leq C \big( 1 + | x | \big);</math>
:<math>\big| \mu (x, t) - \mu (y, t) \big| + \big| \sigma (x, t) - \sigma (y, t) \big| \leq D | x - y |;</math>

for all ''t''&nbsp;∈&nbsp;[0,&nbsp;''T''] and all ''x'' and ''y''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>, where

:<math>| \sigma |^{2} = \sum_{i, j = 1}^{n} | \sigma_{ij} |^{2}.</math>

Let ''Z'' be a random variable that is independent of the ''σ''-algebra generated by ''B''<sub>''s''</sub>, ''s''&nbsp;≥&nbsp;0, and with finite [[moment (mathematics)|second moment]]:

:<math>\mathbb{E} \big[ | Z |^{2} \big] < + \infty.</math>

Then the stochastic differential equation/initial value problem

:<math>\mathrm{d} X_{t} = \mu (X_{t}, t) \, \mathrm{d} t + \sigma (X_{t}, t) \, \mathrm{d} B_{t} \mbox{ for } t \in [0, T];</math>
:<math>X_{0} = Z;</math>

has a P-[[almost surely]] unique ''t''-continuous solution (''t'',&nbsp;''ω'')&nbsp;↦&nbsp;''X''<sub>''t''</sub>(''ω'') such that ''X'' is [[adapted process|adapted]] to the [[filtration (abstract algebra)|filtration]] ''F''<sub>''t''</sub><sup>''Z''</sup> generated by ''Z'' and ''B''<sub>''s''</sub>, ''s''&nbsp;≤&nbsp;''t'', and

:<math>\mathbb{E} \left[ \int_{0}^{T} | X_{t} |^{2} \, \mathrm{d} t \right] < + \infty.</math>

=== General case: local Lipschitz condition and maximal solutions ===
The stochastic differential equation above is only a special case of a more general form
:<math>\mathrm{d}Y_t=\alpha(t,Y_t)\mathrm{d}X_t</math>
where
* <math>X</math> is a continuous semimartingale in <math>\R^n</math> and  <math>Y</math> is a continuous semimartingal in <math>\R^d</math>
* <math>\alpha:\R_{+}\times U \to \operatorname{Lin}(\R^{n};\R^{d})</math> is a map from some open nonempty set <math>U\subset \R^d</math>, where <math>\operatorname{Lin}(\R^{n};\R^{d})</math> is the space of all linear maps from <math>\R^{n}</math> to <math>\R^{d}</math>.
More generally one can also look at stochastic differential equations on [[manifold]]s.

Whether the solution of this equation explodes depends on the choice of <math>\alpha</math>. Suppose <math>\alpha</math> satisfies some local Lipschitz condition, i.e., for <math>t\geq 0</math> and some compact set <math>K\subset U</math> and some constant <math>L(t,K)</math> the condition
:<math>|\alpha(s,y)-\alpha(s,x)|\leq L(t,K)|y-x|,\quad x,y\in K,\;0\leq s\leq t,</math>
where <math>|\cdot|</math> is the Euclidean norm. This condition guarantees the existence and uniqueness of a so-called ''maximal solution''.

Suppose <math>\alpha</math> is continuous and satisfies the above local Lipschitz condition and let <math>F:\Omega\to U</math> be some initial condition, meaning it is a measurable function with respect to the initial σ-algebra. Let <math>\zeta:\Omega\to \overline{\R}_{+}</math> be a [[predictable stopping time]] with <math>\zeta>0</math> almost surely. A <math>U</math>-valued semimartingale <math>(Y_t)_{t<\zeta}</math> is called a ''maximal solution'' of
:<math>dY_t=\alpha(t,Y_t)dX_t,\quad Y_0=F</math>
with ''life time'' <math>\zeta</math> if
* for one (and hence all) announcing <math>\zeta_n\nearrow\zeta</math> the stopped process <math>Y^{\zeta_n}</math> is a solution to the ''stopped stochastic differential equation''
::<math>\mathrm{d}Y=\alpha(t,Y)\mathrm{d}X^{\zeta_n}</math>
* on the set <math>\{\zeta <\infty\}</math> we have almost surely that <math>Y_{t}\to\partial U</math> with <math>t\to \zeta</math>.<ref>{{cite book|first1=Wolfgang|last1=Hackenbroch|first2=Anton|last2=Thalmaier|publisher=Vieweg+Teubner Verlag Wiesbaden|title=Stochastische Analysis: Eine Einführung in die Theorie der stetigen Semimartingale|date=1994 |isbn=978-3-519-02229-9|pages=297–299|lang=de}}</ref>
<math>\zeta</math> is also a so-called ''explosion time''.

==Some explicitly solvable examples==

Explicitly solvable SDEs include:<ref name="kloeden"/>
 
===Linear SDE: General case===
:<math>\mathrm{d}X_t=(a(t)X_t+c(t))\mathrm{d}t+(b(t)X_t+d(t))\mathrm{d}W_t</math>
:<math>X_t=\Phi_{t,t_0}\left(X_{t_0}+\int_{t_0}^t\Phi^{-1}_{s,t_0}(c(s)-b(s)d(s))\mathrm{d}s+\int_{t_0}^t\Phi^{-1}_{s,t_0}d(s)\mathrm{d}W_s\right)</math>
where
:<math>\Phi_{t,t_0}=\exp\left(\int_{t_0}^t\left(a(s)-\frac{b^2(s)}{2}\right)\mathrm{d}s+\int_{t_0}^tb(s)\mathrm{d}W_s\right)</math>

===Reducible SDEs: Case 1===
:<math>\mathrm{d}X_t=\frac12f(X_t)f'(X_t)\mathrm{d}t+f(X_t)\mathrm{d}W_t</math>
for a given differentiable function <math>f</math> is equivalent to the Stratonovich SDE
:<math>\mathrm{d}X_t=f(X_t)\circ W_t</math>
which has a general solution
:<math>X_t=h^{-1}(W_t+h(X_0))</math>
where
:<math>h(x)=\int^{x}\frac{\mathrm{d}s}{f(s)}</math>

===Reducible SDEs: Case 2===
:<math>\mathrm{d}X_t=\left(\alpha f(X_t)+\frac12 f(X_t)f'(X_t)\right)\mathrm{d}t+f(X_t)\mathrm{d}W_t</math>
for a given differentiable function <math>f</math> is equivalent to the Stratonovich SDE
:<math>\mathrm{d}X_t=\alpha f(X_t)\mathrm{d}t + f(X_t)\circ W_t</math>
which is reducible to
:<math>\mathrm{d}Y_t=\alpha \mathrm{d}t+\mathrm{d}W_t</math>
where <math>Y_t=h(X_t)</math> where <math>h</math> is defined as before.
Its general solution is
:<math>X_t=h^{-1}(\alpha t+W_t+h(X_0))</math>

== SDEs and supersymmetry ==
{{Main|Supersymmetric theory of stochastic dynamics}}

In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the [[differential form]]s on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological [[supersymmetry]] which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as [[Chaos theory|chaos]], [[turbulence]], [[self-organized criticality]] etc. and the [[Goldstone theorem]] explains the associated long-range dynamical behavior, i.e., [[Butterfly effect|the butterfly effect]], [[Pink noise|1/f]] and [[Crackling noise|crackling]] noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.

==See also==
* [[Backward stochastic differential equation]]
* [[Langevin dynamics]]
* [[Local volatility]]
* [[Stochastic process]]
* [[Stochastic volatility]]
* [[Stochastic partial differential equations]]
* [[Diffusion process]]
* [[Stochastic difference equation]]

==References==
{{Reflist}}

==Further reading==
* Evans, Lawrence C (2013). [https://bookstore.ams.org/mbk-82 An Introduction to Stochastic Differential Equations] American Mathematical Society.
* {{cite book
| last = Adomian
| first = George
| title = Stochastic systems
| series = Mathematics in Science and Engineering (169)
| publisher = Academic Press Inc.
| location = Orlando, FL
| year = 1983
}}
* {{cite book
| last = Adomian
| first = George
| title = Nonlinear stochastic operator equations
| url = https://archive.org/details/nonlinearstochas0000adom
| url-access = registration
| publisher = Academic Press Inc.
| location = Orlando, FL
| year = 1986
| isbn = 978-0-12-044375-8
}}
* {{cite book
| last = Adomian
| first = George
| title = Nonlinear stochastic systems theory and applications to physics
| series = Mathematics and its Applications (46)
| publisher = Kluwer Academic Publishers Group
| location = Dordrecht
| year = 1989
}}
* {{cite book|last=Calin|first=Ovidiu|title=An Informal Introduction to Stochastic Calculus with Applications|publisher=World Scientific Publishing|location=Singapore|year=2015|isbn=978-981-4678-93-3|page=315}}
* {{cite book
| editor1-last = Teugels | editor1-first = J.
| editor2-last = Sund | editor2-first = B.
| title = Encyclopedia of Actuarial Science
| publisher = Wiley
| location = Chichester
| year = 2004
| pages = 523&ndash;527
}}
* {{cite book
| author = C. W. Gardiner
| title = Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences
| publisher = Springer
| year = 2004
| page = 415
| author-link = Crispin Gardiner
}}
* {{cite book
| author = Thomas Mikosch
| title = Elementary Stochastic Calculus: with Finance in View
| publisher = World Scientific Publishing
| location = Singapore
| year = 1998
|isbn = 981-02-3543-7
| page = 212
}}
* {{cite journal
| author = Seifedine Kadry
| title = A Solution of Linear Stochastic Differential Equation
| journal = Wseas Transactions on Mathematics
| publisher = WSEAS TRANSACTIONS on MATHEMATICS, April 2007.
| location = USA
| year = 2007
|issn = 1109-2769
| page = 618
}}
* {{cite journal|last1=Higham.|first1=Desmond J.|title=An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations|journal=SIAM Review|date=January 2001|volume=43|issue=3|pages=525–546|doi=10.1137/S0036144500378302|bibcode=2001SIAMR..43..525H|citeseerx=10.1.1.137.6375}}
* Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM, {{ISBN|978-1-611976-42-7}} (2021).

{{Stochastic processes}}
{{Authority control}}

[[Category:Stochastic differential equations| ]]
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[[Category:Stochastic processes]]
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