Doléans-Dade exponential ($\mathcal{E}(X)$)

For a continuous semimartingale $X$, its Doléans-Dade exponential isIt solves $d{Z}_t=Z_{t}d{X}_t$ with $Z_0=1$.

If a continuous local martingale $M$ satisfiesthen its stochastic exponential is a true martingale through time $T$.

Girsanov's theorem describes how a change of probability measure changes the drift of a semimartingale. If the appropriate stochastic exponential is a true martingale, weighting by it turns $W_t+{\int }_0^t{\theta }_{s}ds$ into a Brownian motion under the new measure.