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 dY_t = Y_t dX_t with initial condition Y₀ = 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 − X₀). 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

$\begin{align} d\log(Y) &= \frac{1}{Y}\,dY -\frac{1}{2Y^2}\,d[Y] \\ &= dX - \frac{1}{2}\,d[X]. \end{align}$

Exponentiating gives the solution

$Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr),\qquad t\ge0.$

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 (uniformly integrable) 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

$Y_t = \exp\Bigl(X_t-X_0-\frac12[X]_t\Bigr)\prod_{s\le t}(1+\Delta X_s) \exp \Bigl(-\Delta X_s+\frac12\Delta X_s^2\Bigr),\qquad t\ge0,$

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

See also

• Stochastic logarithm

References

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