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→Equivalent definitions: explained/linked "(unweighted) distance"
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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 19:58, 16 February 2014

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