Doléans-Dade exponential

Summary

The Doléans-Dade exponential, or stochastic exponential, $\mathcal{E}(X)$ of a semimartingale $X$ is the solution of the equation $dZ = Z \, dX$ with $Z_0 = 1$. It plays the role of the exponential function in Itô calculus. For a continuous semimartingale with $X_0 = 0$ it equals $\exp(X - \tfrac12 [X])$. It is the density process in the Girsanov theorem.