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 *dY _{t}* =

*Y*with initial condition

_{t}dX_{t}*Y*

_{0}= 1. The concept is named after Catherine Doléans-Dade. It is sometimes denoted by

*Ɛ*(

*X*). In the case where

*X*is differentiable, then

*Y*is given by the differential equation

*dY*/

*dt*=

*Y dX*/

*dt*to which the solution is

*Y*= exp(

*X*−

*X*

_{0}). Alternatively, if

*X*=

_{t}*σB*+

_{t}*μt*for a Brownian motion

*B*, then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale

*X*, applying Itō's lemma with

*ƒ*(

*Y*) = log(

*Y*) gives

Exponentiating gives the solution

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 [*X*] in the solution.

The Doléans-Dade exponential is useful in the case when *X* is a local martingale. Then, *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 *Ɛ*(*X*) is actually a 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

where the product extents over the (countable many) jumps of *X* up to time *t*.

## See also[edit]

## References[edit]

- Protter, Philip E. (2004),
*Stochastic Integration and Differential Equations*(2nd ed.), Springer, ISBN 3-540-00313-4

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