|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Schilder's theorem''' is a result in the [[large deviations theory]] of [[stochastic process]]es. Roughly speaking, Schilder's theorem gives an estimate for the probability that a (scaled-down) sample path of [[Brownian motion]] will stray far from the mean path (which is constant with value 0). This statement is made precise using [[rate function]]s. Schilder's theorem is generalized by the [[Freidlin–Wentzell theorem]] for [[Itō diffusion]]s.
| | Hi there. My title is Sophia Meagher even though it is not the title on my beginning certification. To perform lacross is some thing he would by no means give up. Some time in the past best psychic; [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review clothingcarearchworth.com], she selected to live in Alaska and her parents reside close by. Credit authorising is exactly where my primary income arrives from.<br><br>My page ... [http://m-card.co.kr/xe/mcard_2013_promote01/29877 cheap psychic readings] readers ([http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow]) |
| | |
| ==Statement of the theorem==
| |
| Let ''B'' be a standard Brownian motion in ''d''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''d''</sup> starting at the origin, 0 ∈ '''R'''<sup>''d''</sup>; let '''W''' denote the [[law (stochastic processes)|law]] of ''B'', i.e. classical [[Wiener measure]]. For ''ε'' > 0, let '''W'''<sub>''ε''</sub> denote the law of the rescaled process (√''ε'')''B''. Then, on the [[Banach space]] ''C''<sub>0</sub> = ''C''<sub>0</sub>([0, ''T'']; '''R'''<sup>''d''</sup>) of continuous functions <math> f : [0,T] \longrightarrow \mathbf{R}^d</math> such that <math>f(0)=0</math>, equipped with the [[supremum norm]] ||·||<sub>∞</sub>, the [[probability measure]]s '''W'''<sub>''ε''</sub> satisfy the large deviations principle with good rate function ''I'' : ''C''<sub>0</sub> → '''R''' ∪ {+∞} given by
| |
| | |
| :<math>I(\omega) = \frac{1}{2} \int_{0}^{T} | \dot{\omega}(t) |^{2} \, \mathrm{d} t</math>
| |
| | |
| if ''ω'' is [[absolutely continuous]], and ''I''(''ω'') = +∞ otherwise. In other words, for every [[open set]] ''G'' ⊆ ''C''<sub>0</sub> and every [[closed set]] ''F'' ⊆ ''C''<sub>0</sub>,
| |
| | |
| :<math>\limsup_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{W}_{\varepsilon} (F) \leq - \inf_{\omega \in F} I(\omega)</math>
| |
| | |
| and | |
| | |
| :<math>\liminf_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{W}_{\varepsilon} (G) \geq - \inf_{\omega \in G} I(\omega).</math>
| |
| | |
| ==Example==
| |
| Taking ''ε'' = 1 ⁄ ''c''<sup>2</sup>, one can use Schilder's theorem to obtain estimates for the probability that a standard Brownian motion ''B'' strays further than ''c'' from its starting point over the time interval [0, ''T''], i.e. the probability
| |
| | |
| :<math>\mathbf{W} (C_{0} \setminus \mathbf{B}_{c} (0; \| \cdot \|_{\infty})) \equiv \mathbf{P} \big[ \| B \|_{\infty} > c \big],</math> | |
| | |
| as ''c'' tends to infinity. Here '''B'''<sub>''c''</sub>(0; ||·||<sub>∞</sub>) denotes the [[open ball]] of radius ''c'' about the zero function in ''C''<sub>0</sub>, taken with respect to the [[supremum norm]]. First note that
| |
| | |
| :<math>\| B \|_{\infty} > c \iff \sqrt{\varepsilon} B \in A := \big\{ \omega \in C_{0} \big| | \omega(t) | > 1 \mbox{ for some } t \in [0, T] \big\}.</math>
| |
| | |
| Since the rate function is continuous on ''A'', Schilder's theorem yields
| |
| | |
| :<math>\lim_{c \to \infty} \frac{1}{c^{2}} \log \mathbf{P} \big[ \| B \|_{\infty} > c \big]</math> | |
| ::<math>= \lim_{\varepsilon \to 0} \epsilon \mathbf{P} \big[ \sqrt{\varepsilon} B \in A \big]</math>
| |
| ::<math>= - \inf \left\{ \left. \frac{1}{2} \int_{0}^{T} | \dot{\omega}(t) |^{2} \, \mathrm{d} t \right| \omega \in A \right\}</math>
| |
| ::<math>= - \frac{1}{2} \int_{0}^{T} \frac{1}{T^{2}} \, \mathrm{d} t</math>
| |
| ::<math>= - \frac{1}{2 T},</math>
| |
| | |
| making use of the fact that the [[infimum]] over paths in the collection ''A'' is attained for ''ω''(''t'') = ''t'' ⁄ ''T''. This result can be heuristically interpreted as saying that, for large ''c'' and/or large ''T''
| |
| | |
| :<math>\frac{1}{c^{2}} \log \mathbf{P} \big[ \| B \|_{\infty} > c \big] \approx - \frac{1}{2 T},</math> | |
| | |
| or, in other words,
| |
| | |
| :<math>\mathbf{P} \big[ \| B \|_{\infty} > c \big] \approx \exp \left( - \frac{c^{2}}{2 T} \right).</math>
| |
| | |
| In fact, the above probability can be estimated more precisely as follows: for ''B'' a standard Brownian motion in '''R'''<sup>''n''</sup>, and any ''T'', ''c'' and ''ε'' > 0, it holds that
| |
| | |
| :<math>\mathbf{P} \left[ \sup_{0 \leq t \leq T} \big| \sqrt{\varepsilon} B_{t} \big| \geq c \right] \leq 4 n \exp \left( - \frac{c^{2}}{2 n T \varepsilon} \right).</math>
| |
| | |
| ==References==
| |
| * {{cite book
| |
| | last= Dembo
| |
| | first = Amir
| |
| | coauthors = Zeitouni, Ofer
| |
| | title = Large deviations techniques and applications
| |
| | series = Applications of Mathematics (New York) 38
| |
| | edition = Second edition
| |
| | publisher = Springer-Verlag
| |
| | location = New York
| |
| | year = 1998
| |
| | pages = xvi+396
| |
| | isbn = 0-387-98406-2
| |
| | MR=1619036}} (See theorem 5.2)
| |
| | |
| [[Category:Asymptotic analysis]]
| |
| [[Category:Stochastic processes]]
| |
| [[Category:Probability theorems]]
| |
| [[Category:Large deviations theory]]
| |
Hi there. My title is Sophia Meagher even though it is not the title on my beginning certification. To perform lacross is some thing he would by no means give up. Some time in the past best psychic; clothingcarearchworth.com, she selected to live in Alaska and her parents reside close by. Credit authorising is exactly where my primary income arrives from.
My page ... cheap psychic readings readers (http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow)