Doléans-Dade exponential

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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 dYt = Yt− dXt 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 = YdX/dt to which the solution is Y = exp(X-X0). Alternatively, if X = σB + 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 f(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\left(X_t-X_0-[X]_t/2\right).\,

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.

It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans exponential of any semimartingale X is

Y_t=\exp\left(X_t-X_0-[X]_t/2\right)\prod_{s\le t}(1+\Delta X_s)\exp\left(-\Delta X_s+\Delta X_s^2/2\right).

[edit] References

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