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en>Jochen Burghardt
→‎Equivalent definitions: explained/linked "(unweighted) distance"
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Undid revision 595729391 by Jochen Burghardt (talk) gratuitous and out-of-concensus format change to references (CS1 instead of CS2), breaks harv referencing
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In [[stochastic calculus]], the '''Doléans-Dade exponential''', '''Doléans exponential''', or '''stochastic exponential''', of a [[semimartingale]] ''X'' is defined to be the solution to the [[stochastic differential equation]]  {{nowrap|''dY<sub>t</sub>'' {{=}} ''Y<sub>t</sub> dX<sub>t</sub>''}} with initial condition  {{nowrap|''Y''<sub>0</sub> {{=}} 1}}. The concept is named after [[Catherine Doléans-Dade]]. It is sometimes denoted by ''{{unicode|Ɛ}}''(''X'').
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In the case where ''X'' is differentiable, then ''Y'' is given by the differential equation  {{nowrap|''dY''/''dt'' {{=}} ''Y dX''/''dt''}} to which the solution is  {{nowrap|''Y'' {{=}} exp(''X'' &minus; ''X''<sub>0</sub>)}}.
Alternatively, if  {{nowrap|''X<sub>t</sub>'' {{=}} ''σB<sub>t</sub>'' + ''μt''}} for a [[Wiener process|Brownian motion]] ''B'', then the Doléans-Dade exponential is a [[geometric Brownian motion]]. For any continuous semimartingale ''X'', applying [[Itō's lemma]] with  {{nowrap|''&fnof;''&thinsp;(''Y'') {{=}} log(''Y'')}} gives


:<math>
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\begin{align}
d\log(Y) &= \frac{1}{Y}\,dY -\frac{1}{2Y^2}\,d[Y] \\
&= dX - \frac{1}{2}\,d[X].
\end{align}
</math>
 
Exponentiating gives the solution
 
:<math>Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr),\qquad t\ge0.</math>
 
This differs from what might be expected by comparison with the case where ''X'' is differentiable due to the existence of the [[quadratic variation]] term  <nowiki>[</nowiki>''X''<nowiki>]</nowiki> in the solution.
 
The Doléans-Dade exponential is useful in the case when ''X'' is a [[local martingale]]. Then, ''{{unicode|Ɛ}}''(''X'') will also be a local martingale whereas the normal exponential exp(''X'') is not. This is used in the [[Girsanov theorem]]. Criteria for a continuous local martingale ''X'' to ensure that its stochastic exponential ''{{unicode|Ɛ}}''(''X'') is actually a [[martingale (probability theory)|martingale]] are given by [[Kazamaki's condition]], [[Novikov's condition]] and [[Beneš' condition]].
 
It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale ''X'' is
 
:<math>Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr)\prod_{s\le t}(1+\Delta X_s) \exp \Bigl(-\Delta X_s+\frac12\Delta X_s^2\Bigr),\qquad t\ge0,</math>
 
where the product extents over the (countable many) jumps of ''X'' up to time ''t''.
 
== See also ==
* [[Stochastic logarithm]]
 
== References ==
 
*{{Citation|last=Protter|first=Philip E.|year=2004|title=Stochastic Integration and Differential Equations|publisher=Springer|edition=2nd|isbn=3-540-00313-4}}
 
{{DEFAULTSORT:Doleans-Dade exponential}}
[[Category:Martingale theory]]
[[Category:Stochastic differential equations]]

Revision as of 20:58, 16 February 2014

By investing in a premium Word - Press theme, you're investing in the future of your website. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. Keep reading for some great Word - Press ideas you can start using today. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.

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