Doléans-Dade exponential
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 with initial condition . The concept is named after Catherine Doléans-Dade. It is sometimes denoted by Ɛ.
In the case where X is differentiable, then Y is given by the differential equation to which the solution is .
Alternatively, if 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 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 in the solution.
The Doléans-Dade exponential is useful in the case when X is a local martingale. Then, Ɛ will also be a local martingale whereas the normal exponential exp is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale X to ensure that its stochastic exponential Ɛ is actually a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's 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 jumps of X up to time t.
