In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation $${\displaystyle dY_{t}=Y_{t-}\,dX_{t},\quad \quad Y_{0}=1,}$$where ${\displaystyle Y_{-}}$ denotes the process of left limits, i.e., ${\displaystyle Y_{t-}=\lim _{s\uparrow t}Y_{s}}$.

The concept is named after Catherine Doléans-Dade. The stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since ${\displaystyle X}$ measures the cumulative percentage change in ${\displaystyle Y}$.

Process ${\displaystyle Y}$ obtained above is commonly denoted by ${\displaystyle {\mathcal {E}}(X)}$. The terminology "stochastic exponential" arises from the similarity of ${\displaystyle {\mathcal {E}}(X)=Y}$ to the natural exponential of ${\displaystyle X}$: If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation ${\displaystyle dY_{t}/\mathrm {d} t=Y_{t}dX_{t}/dt}$, whose solution is ${\displaystyle Y=\exp(X-X_{0})}$.

Yor's formula: for any two semimartingales ${\displaystyle U}$ and ${\displaystyle V}$ one has $${\displaystyle {\mathcal {E}}(U){\mathcal {E}}(V)={\mathcal {E}}(U+V+[U,V])}$$

For any continuous semimartingale X, take for granted that ${\displaystyle Y}$ is continuous and strictly positive. Then applying Itō's formula with ƒ(Y) = log(Y) gives

Exponentiating with ${\displaystyle Y_{0}=1}$ gives the solution

This differs from what might be expected by comparison with the case where X has finite variation due to the existence of the quadratic variation term [X] in the solution.