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| In the [[stochastic calculus]], '''Tanaka's formula''' states that
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| :<math>|B_t| = \int_0^t \sgn(B_s)\, dB_s + L_t</math>
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| where ''B''<sub>''t''</sub> is the standard [[Brownian motion]], sgn denotes the [[sign function]]
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| :<math>\sgn (x) = \begin{cases} +1, & x \geq 0; \\ -1, & x < 0. \end{cases}</math>
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| and ''L''<sub>''t''</sub> is its [[Local time (mathematics)|local time]] at 0 (the local time spent by ''B'' at 0 before time ''t'') given by the [[Lp space|''L''<sup>2</sup>-limit]]
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| :<math>L_{t} = \lim_{\varepsilon \downarrow 0} \frac1{2 \varepsilon} | \{ s \in [0, t] | B_{s} \in (- \varepsilon, + \varepsilon) \} |.</math>
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| Tanaka's formula is the explicit [[Doob–Meyer decomposition theorem|Doob–Meyer decomposition]] of the submartingale |''B''<sub>''t''</sub>| into the [[martingale (probability theory)|martingale]] part (the [[integral]] on the right-hand side), and a continuous increasing process (local time). It can also be seen as the analogue of [[Itō's lemma]] for the (nonsmooth) absolute value function <math>f(x)=|x|</math>, with <math> f'(x) = \sgn(x)</math> and <math> f''(x) = 2\delta(x) </math>; see [[Local time (mathematics)|local time]] for a formal explanation of the Itō term.
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| == Outline of proof ==
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| The [[Function (mathematics)|function]] |''x''| is not [[smooth function|''C''<sup>2</sup>]] in ''x'' at ''x'' = 0, so we cannot apply [[Itō's formula]] directly. But if we approximate it near zero (i.e. in [−''ε'', ''ε'']) by [[parabola]]s
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| :<math>\frac{x^2}{2|\varepsilon|}+\frac{|\varepsilon|}{2}.</math>
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| And using [[Itō's formula]] we can then take the [[Limit (mathematics)|limit]] as ''ε'' → 0, leading to Tanaka's formula.
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| == References ==
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| * {{cite book
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| | last = Øksendal
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| | first = Bernt K.
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| | authorlink = Bernt Øksendal
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| | title = Stochastic Differential Equations: An Introduction with Applications
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| | edition = Sixth edition
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| | publisher=Springer
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| | location = Berlin
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| | year = 2003
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| | isbn = 3-540-04758-1
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| }} (Example 5.3.2)
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| * {{cite book
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| | last = Shiryaev
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| | first = Albert N.
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| |authorlink= Albert Shiryaev
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| | title = Essentials of stochastic finance: Facts, models, theory
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| | series = Advanced Series on Statistical Science & Applied Probability No. 3
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| |coauthors = trans. N. Kruzhilin
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| |publisher = World Scientific Publishing Co. Inc.
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| | location = River Edge, NJ
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| | year = 1999
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| | isbn = 981-02-3605-0
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| }}
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| [[Category:Equations]]
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| [[Category:Martingale theory]]
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| [[Category:Probability theorems]]
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The name of the writer is Numbers. For many years I've been working as a payroll clerk. Minnesota is exactly where he's been living for years. He is really fond of doing ceramics but he is struggling to find time for it.
Feel free to surf to my web page :: std testing at home