Editing Stochastic differential equation (section)

=== 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"/>