en>Mark viking: Added context, bolded title, added wl, c/e

2013-11-15T06:15:08Z

Added context, bolded title, added wl, c/e

← Older revision		Revision as of 08:15, 15 November 2013
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	~~Andrew Simcox~~ is the ~~name his mothers and fathers gave him and he completely loves this name~~. It's ~~not a common thing but what I like performing~~ is ~~to climb but I don~~'~~t have~~ the ~~time recently. Alaska~~ [~~http~~:/~~/www~~.~~khuplaza~~.~~com/dent/14869889 psychic readers]~~ is the ~~only location I~~'~~ve been residing in~~ [~~http://www.sirudang.com/siroo_Notice/2110 love psychic readings~~] ~~but now I~~'~~m considering other choices~~. ~~Since I was 18 I~~'~~ve been working~~ as a ~~bookkeeper but quickly my wife~~ and ~~I will start our own company.~~<br><br>~~Feel free to surf to my web page~~: ~~[http~~://~~isaworld~~.pe.~~kr/?document_srl~~=~~392088 tarot readings~~]		In [[mathematics]], '''Tanaka's equation''' is an example of a [[stochastic differential equation]] which admits a weak solution but has no strong solution. It is named after the [[Japan]]ese [[mathematician]] Hiroshi Tanaka.

			Tanaka's equation is the one-dimensional stochastic differential equation

			:<math>\mathrm{d} X_t = \sgn (X_t) \, \mathrm{d} B_t,</math>

			driven by canonical [[Brownian motion]] ''B'', with initial condition ''X''<sub>0</sub> = 0, where sgn denotes the [[sign function]]

			:<math>\sgn (x) = \begin{cases} +1, & x \geq 0; \\ -1, & x < 0. \end{cases}</math>

			(Note the unconventional value for sgn(0).) The signum function does not satisfy the [[Lipschitz continuity]] condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. In fact, the Tanaka equation has no strong solution, i.e. one for which the version ''B'' of Brownian motion is given in advance and the solution ''X'' is [[adapted process\|adapted]] to the [[filtration (abstract algebra)\|filtration]] generated by ''B'' and the initial conditions. However, the Tanaka equation does have a weak solution, one for which the process ''X'' and version of Brownian motion are both specified as part of the solution, rather than the Brownian motion being given ''[[prior probability\|a priori]]''. In this case, simply choose ''X'' to be any Brownian motion <math>\hat{B}</math> and define <math>\tilde{B}</math> by

			:<math>\tilde{B}_t = \int_0^t \sgn \big( \hat{B}_s \big) \, \mathrm{d} \hat{B}_s = \int_0^t \sgn \big( X_s \big) \, \mathrm{d} X_s,</math>

			i.e.

			:<math>\mathrm{d} \tilde{B}_t = \sgn (X_t) \, \mathrm{d} X_t.</math>

			Hence,

			:<math>\mathrm{d} X_t = \sgn (X_t) \, \mathrm{d} \tilde{B}_{t},</math>

			and so ''X'' is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same [[law (stochastic processes)\|law]].

			==References==

			* {{cite book
			\| last = Øksendal
			\| first = Bernt K.
			\| authorlink = Bernt Øksendal
			\| title = Stochastic Differential Equations: An Introduction with Applications
			\| edition = Sixth edition
			\| publisher=Springer
			\| location = Berlin
			\| year = 2003
			\| id = ISBN 3-540-04758-1
			}} (Example 5.3.2)

			[[Category:Equations]]
			[[Category:Stochastic differential equations]]

en>Ohconfucius: General formatting, inc. removal of superscripted ordinals

2011-10-19T14:26:48Z

General formatting, inc. removal of superscripted ordinals

New page

Andrew Simcox is the name his mothers and fathers gave him and he completely loves this name. It's not a common thing but what I like performing is to climb but I don't have the time recently. Alaska [http://www.khuplaza.com/dent/14869889 psychic readers] is the only location I've been residing in [http://www.sirudang.com/siroo_Notice/2110 love psychic readings] but now I'm considering other choices. Since I was 18 I've been working as a bookkeeper but quickly my wife and I will start our own company.<br><br>Feel free to surf to my web page: [http://isaworld.pe.kr/?document_srl=392088 tarot readings]

Legendre wavelet - Revision history

en>Mark viking: Added context, bolded title, added wl, c/e

en>Ohconfucius: General formatting, inc. removal of superscripted ordinals