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Stochastic differential equation
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==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'' > 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'' ∈ [0, ''T''] and all ''x'' and ''y'' ∈ '''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'' ≥ 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'', ''ω'') ↦ ''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'' ≤ ''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''.
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