Doléans-Dade exponential

Remarks

If $X$ is continuous, then $\mathcal{E}(X)_t = \exp\left(X_t - X_0 - \tfrac12 [X]_t\right)$, where $[X]$ is the quadratic variation of $X$. In general, $\mathcal{E}(X)_t = \exp\left(X_t - X_0 - \tfrac12 [X]^c_t\right) \prod_{0 < s \le t} (1 + \Delta X_s) e^{-\Delta X_s}$, where $[X]^c$ is the continuous part of $[X]$ and $\Delta X_s = X_s - X_{s-}$. Example: let $B$ be a Brownian motion and $X = \sigma B$ with a constant $\sigma$. Then $Z_t = \exp(\sigma B_t - \tfrac12 \sigma^2 t)$, and Itô's lemma gives $dZ_t = \sigma Z_t \, dB_t = Z_t \, dX_t$. Non-example: $W_t = \exp(\sigma B_t)$ fails the equation, since Itô's lemma gives $dW_t = W_t \, dX_t + \tfrac12 \sigma^2 W_t \, dt$. If $X$ is a continuous local martingale, then $\mathcal{E}(X)$ is a positive local martingale and hence a supermartingale. It is a true martingale under Novikov's condition $E\left[\exp\left(\tfrac12 [X]_\infty\right)\right] < \infty$.