Editing Stochastic differential equation (section)

==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"/>