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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>
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| ==Description==
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| Consider the [[autonomous equation|autonomous]] [[Itō calculus|Itō]] stochastic differential equation
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| :<math>\mathrm{d} X_t = a(X_t) \, \mathrm{d} t + b(X_t) \, \mathrm{d} W_t,</math>
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| 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:
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| * partition the interval [0, ''T''] into ''N'' equal subintervals of width <math>\Delta t>0</math>:
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| :<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>
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| * set <math>Y_0 = x_0;</math>
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| * recursively define <math>Y_n</math> for <math>1 \leq n \leq N</math> by
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| :<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>
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| where <math>b'</math> denotes the [[derivative]] of <math>b(x)</math> with respect to <math>x</math> and
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| :<math>\Delta W_n = W_{\tau_{n + 1}} - W_{\tau_n}</math> | |
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| 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.
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| 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>
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| == Intuitive derivation ==
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| For this derivation, we will only look at [[geometric Brownian motion]] (GBM), the stochastic differential equation of which is given by
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| :<math>\mathrm{d} X_t = \mu X \mathrm{d} t + \sigma X d W_t</math>
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| with real constants <math>\mu</math> and <math>\sigma</math>. Using [[Itō's lemma]] we get
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| :<math>\mathrm{d}\ln X_t=\left(\mu-\frac{1}{2}\sigma^2\right)\mathrm{d}t+\sigma\mathrm{d}W_t,</math>
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| Thus, the solution to the GBM SDE is
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| :<math>
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| \begin{align}
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| 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\} \\
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| &\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) \\
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| &= 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)
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| \end{align}
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| </math>
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| where
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| :<math> a(x) = \mu x, ~b(x) = \sigma x </math>.
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| == See also ==
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| * [[Euler–Maruyama method]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * {{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 }}
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| [[Category:Numerical differential equations]]
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| [[Category:Stochastic differential equations]]
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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.
Check out my website; http://www.eddysadventurestore.nl