Commutation matrix - Revision history

128.62.24.174: commutation matrix is involution and symmetric only in special case m = n; mention this and general relation

2014-06-27T21:19:36Z

commutation matrix is involution and symmetric only in special case m = n; mention this and general relation

← Older revision		Revision as of 23:19, 27 June 2014
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	~~In [[mathematics]],~~ the ~~'''Milstein method'''~~ is ~~a technique for the approximate [[numerical analysis\|numerical solution]] of a [[stochastic differential equation]]~~. It is ~~named after [[Grigori N. Milstein]] who first published the method in 1974.<ref>{{cite journal\|first=G. N.\|last=Mil~~'~~shtein\|title=Approximate integration of stochastic differential equations\|journal=Teor~~. Veroyatnost. i Primenen\|volume=19\|issue=3\|year=1974\| pages=583–588\| url =http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=2929&option_lang=eng\|language=Russian}}</ref><ref>{{cite doi\|10.1137/1119062}}</ref>		The title of the writer is Figures. She is a librarian but she's always needed her own company. Doing ceramics is what her family members and her enjoy. North Dakota is our beginning location.<br><br>Check out my website; [http://www.eddysadventurestore.nl/nieuws/eliminate-candida-one-these-tips http://www.eddysadventurestore.nl]

	~~==Description==~~
	~~Consider the [[autonomous equation\|autonomous]] [[Itō calculus\|Itō]] stochastic differential equation~~

	~~:<math>\mathrm{d} X_t = a(X_t) \, \mathrm{d} t + b(X_t) \, \mathrm{d} W_t,</math>~~

	~~with [[initial condition]] ''X''<sub>0</sub> = ''x''<sub>0</sub>, where ''W''<sub>''t''</sub> stands for the [[Wiener process]],~~ and ~~suppose that we wish to solve this SDE on some interval of time [0, ''T'']~~. ~~Then the '''Milstein approximation''' to the true solution ''X''~~ is ~~the [[Markov chain]] ''Y'' defined as follows:~~

	* partition the interval [0, ''T''] into ''N'' equal subintervals of width <~~math~~>~~\Delta t>0~~<~~/math~~>:

	~~:<math>0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text{ with }\tau_n:=n\Delta t\text{ and }\Delta t = \frac{T}{N}~~;~~</math>~~

	* set <math>Y_0 = x_0;</math>

	* recursively define <math>Y_n</math> for <math>1 \leq n \leq N</math> by

	~~:<math>Y_{n + 1} = Y_n + a(Y_n) \Delta t + b(Y_n) \Delta W_n + \frac{1}{2} b(Y_n) b'(Y_n) \left( (\Delta W_n)^2 - \Delta t \right),</math>~~

	~~where <math>b'</math> denotes the~~ [~~[derivative]] of <math>b(x)</math> with respect to <math>x</math> and~~

	:~~<math>\Delta W_n = W_{\tau_{n + 1}} - W_{\tau_n}</math>~~

	~~are [[independent and identically distributed]] [[normal distribution\|normal random variables]] with [[expected value]] zero and [[variance]] <math>\Delta t</math>. Then <math>Y_n<~~/~~math> will approximate <math>X_{\tau_n}<~~/~~math> for <math>0 \leq n \leq N</math>, and increasing <math>N</math> will yield a better approximation.~~

	~~The error of the Milstein method is of order <math>\Delta t</math>, which is considerably better than the [[Euler–Maruyama method]], whose error is of order <math>(\Delta t)^{1/2}</math>~~.~~<ref>V~~. ~~Mackevičius, ''Introduction to Stochastic Analysis'', Wiley 2011<~~/~~ref>~~

	~~== Intuitive derivation ==~~

	~~For this derivation, we will only look at [[geometric Brownian motion]] (GBM), the stochastic differential equation of which is given by~~

	~~:<math>\mathrm{d} X_t = \mu X \mathrm{d} t + \sigma X d W_t<~~/~~math>~~

	~~with real constants <math>\mu</math> and <math>\sigma</math>. Using [[Itō's lemma]] we get~~

	~~:<math>\mathrm{d}\ln X_t=\left(\mu~~-~~\frac{1}{2}\sigma^2\right)\mathrm{d}t+\sigma\mathrm{d}W_t,</math>~~
	~~Thus, the solution to the GBM SDE is~~

	~~:<math>~~
	~~\begin{align}~~
	~~X_{t+\Delta t}&=X_t\exp\left\{\int_t^{t+\Delta t}\left(\mu~~-~~\frac{1}{2}\sigma^2\right)\mathrm{d}t+\int_t^{t+\Delta t}\sigma\mathrm{d}W_u\right\} \\~~
	~~&\approx X_t\left(1+\mu\Delta t~~-~~\frac{1}{2}\sigma^2\Delta t+\sigma\Delta W_t+\frac{1}{2}\sigma^2(\Delta W_t)^2\right) \\~~
	~~&= X_t + a(X_t)\Delta t+b(X_t)\Delta W_t+\frac{1}{2}b(X_t)b'(X_t)((\Delta W_t)^2~~-~~\Delta t)~~
	~~\end{align}~~
	</~~math>~~

	~~where~~

	~~:<math> a(x) = \mu x, ~b(x) = \sigma x <~~/~~math>~~.

	~~== See also ==~~
	* [[Euler–Maruyama method]]

	~~==References==~~

	~~{{Reflist}}~~

	~~==Further reading==~~
	* {{cite book \| author=Kloeden, P.~~E., & Platen, E. \| title=Numerical Solution of Stochastic Differential Equations \| publisher=Springer, Berlin \| year=1999 \| isbn=3-540-54062-8 }}~~

	~~[[Category:Numerical differential equations]]~~
	~~[[Category:Stochastic differential equations]~~]

128.174.116.217 at 15:59, 17 December 2013

2013-12-17T15:59:13Z

← Older revision		Revision as of 17:59, 17 December 2013
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	~~The writer~~ is ~~known by~~ [http://~~tomport~~.ru/~~node~~/~~19667 over~~ the ~~counter std test~~] ~~title~~ of ~~Numbers Lint~~. ~~What I adore performing~~ is ~~playing baseball but I haven~~'t ~~made~~ a ~~dime~~ with it. ~~Hiring~~ is ~~my profession~~. ~~South Dakota~~ is where ~~me and my husband live~~.		In [[mathematics]], the '''Milstein method''' is a technique for the approximate [[numerical analysis\|numerical solution]] of a [[stochastic differential equation]]. It is named after [[Grigori N. Milstein]] who first published the method in 1974.<ref>{{cite journal\|first=G. N.\|last=Mil'shtein\|title=Approximate integration of stochastic differential equations\|journal=Teor. Veroyatnost. i Primenen\|volume=19\|issue=3\|year=1974\| pages=583–588\| url =http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=2929&option_lang=eng\|language=Russian}}</ref><ref>{{cite doi\|10.1137/1119062}}</ref>

			==Description==
			Consider the [[autonomous equation\|autonomous]] [[Itō calculus\|Itō]] stochastic differential equation

			:<math>\mathrm{d} X_t = a(X_t) \, \mathrm{d} t + b(X_t) \, \mathrm{d} W_t,</math>

			with [[initial condition]] ''X''<sub>0</sub> = ''x''<sub>0</sub>, where ''W''<sub>''t''</sub> stands for the [[Wiener process]], and suppose that we wish to solve this SDE on some interval of time [0, ''T'']. Then the '''Milstein approximation''' to the true solution ''X'' is the [[Markov chain]] ''Y'' defined as follows:

			* partition the interval [0, ''T''] into ''N'' equal subintervals of width <math>\Delta t>0</math>:

			:<math>0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text{ with }\tau_n:=n\Delta t\text{ and }\Delta t = \frac{T}{N};</math>

			* set <math>Y_0 = x_0;</math>

			* recursively define <math>Y_n</math> for <math>1 \leq n \leq N</math> by

			:<math>Y_{n + 1} = Y_n + a(Y_n) \Delta t + b(Y_n) \Delta W_n + \frac{1}{2} b(Y_n) b'(Y_n) \left( (\Delta W_n)^2 - \Delta t \right),</math>

			where <math>b'</math> denotes the [[derivative]] of <math>b(x)</math> with respect to <math>x</math> and

			:<math>\Delta W_n = W_{\tau_{n + 1}} - W_{\tau_n}</math>

			are [[independent and identically distributed]] [[normal distribution\|normal random variables]] with [[expected value]] zero and [[variance]] <math>\Delta t</math>. Then <math>Y_n</math> will approximate <math>X_{\tau_n}</math> for <math>0 \leq n \leq N</math>, and increasing <math>N</math> will yield a better approximation.

			The error of the Milstein method is of order <math>\Delta t</math>, which is considerably better than the [[Euler–Maruyama method]], whose error is of order <math>(\Delta t)^{1/2}</math>.<ref>V. Mackevičius, ''Introduction to Stochastic Analysis'', Wiley 2011</ref>

			== Intuitive derivation ==

			For this derivation, we will only look at [[geometric Brownian motion]] (GBM), the stochastic differential equation of which is given by

			:<math>\mathrm{d} X_t = \mu X \mathrm{d} t + \sigma X d W_t</math>

			with real constants <math>\mu</math> and <math>\sigma</math>. Using [[Itō's lemma]] we get

			:<math>\mathrm{d}\ln X_t=\left(\mu-\frac{1}{2}\sigma^2\right)\mathrm{d}t+\sigma\mathrm{d}W_t,</math>
			Thus, the solution to the GBM SDE is

			:<math>
			\begin{align}
			X_{t+\Delta t}&=X_t\exp\left\{\int_t^{t+\Delta t}\left(\mu-\frac{1}{2}\sigma^2\right)\mathrm{d}t+\int_t^{t+\Delta t}\sigma\mathrm{d}W_u\right\} \\
			&\approx X_t\left(1+\mu\Delta t-\frac{1}{2}\sigma^2\Delta t+\sigma\Delta W_t+\frac{1}{2}\sigma^2(\Delta W_t)^2\right) \\
			&= X_t + a(X_t)\Delta t+b(X_t)\Delta W_t+\frac{1}{2}b(X_t)b'(X_t)((\Delta W_t)^2-\Delta t)
			\end{align}
			</math>

			where

			:<math> a(x) = \mu x, ~b(x) = \sigma x </math>.

			== See also ==
			* [[Euler–Maruyama method]]

			==References==

			{{Reflist}}

			==Further reading==
			* {{cite book \| author=Kloeden, P.E., & Platen, E. \| title=Numerical Solution of Stochastic Differential Equations \| publisher=Springer, Berlin \| year=1999 \| isbn=3-540-54062-8 }}

			[[Category:Numerical differential equations]]
			[[Category:Stochastic differential equations]]

en>R'n'B: Fix links to disambiguation page Matrix theory

2012-07-27T23:16:50Z

Fix links to disambiguation page Matrix theory

New page

The writer is known by [http://tomport.ru/node/19667 over the counter std test] title of Numbers Lint. What I adore performing is playing baseball but I haven't made a dime with it. Hiring is my profession. South Dakota is where me and my husband live.